Part III

Random Variables and Their Distributions

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Section 8Random Variables[6/83%]
Problem 8.1

Let ξ\xi and η\eta be random variables. Prove that the following sets are events:

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(a)

A={ω∈Ω:ξ(ω)<η(ω)}A = \left\{ \omega \in \Omega : \xi (\omega ) < \eta (\omega )\right\};

(b)

B={ω∈Ω:ξ(ω)⩽η(ω)}B = \left\{ \omega \in \Omega : \xi (\omega ) \leqslant \eta (\omega )\right\};

(c)

C={ω∈Ω:ξ(ω)=η(ω)}C = \left\{ \omega \in \Omega : \xi (\omega ) = \eta (\omega )\right\}.

Problem 8.2

Let the function ξ:Ω→R\xi : \Omega \rightarrow \mathbb {R} be such that for every x∈Rx \in \mathbb {R} the set Ax={ω∈Ω:ξ(ω)=x}A_x = \left\{ \omega \in \Omega : \xi (\omega ) = x\right\} is an event. Is it true that ξ\xi must then be a random variable?

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Problem 8.3

Construct a probability space (Ω,F,P)(\Omega , \mathscr {F}, \mathbb {P}) and a function ξ:Ω→R\xi : \Omega \rightarrow \mathbb {R} on it such that ξ\xi is not a random variable.

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Problem 8.4

Must the function ξ\xi be a random variable if the following function is a random variable:

?
(a)

η=ξ2\eta = \xi^2;

(b)

η=eξ\eta = e^\xi;

(c)

η=∣ξ∣\eta = \left|\xi \right|;

(d)

η=ξ+1\eta = \xi + 1.

Problem 8.5

Let the probability space (Ω,F,P)(\Omega , \mathscr {F}, \mathbb {P}) be the segment [0,1][0, 1] with the σ\sigma-algebra of Borel sets and Lebesgue measure. Describe the σ\sigma-algebra generated by the random variable:

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(a)

ξ(ω)=1/2\xi (\omega ) = 1/2;

(b)

ξ(ω)={ω,ω∈[0,1/2),1,ω∈[1/2,1];\xi (\omega ) = \begin{cases} \omega , & \omega \in [0, 1/2), \\ 1, & \omega \in [1/2, 1]; \end{cases}

(c)

ξ(ω)=ω/2\xi (\omega ) = \omega /2;

(d)

ξ(ω)={1/4,ω∈[0,1/4),1/2,ω∈[1/4,3/4),1,ω∈[3/4,1].\xi (\omega ) = \begin{cases} 1/4, & \omega \in [0, 1/4), \\ 1/2, & \omega \in [1/4, 3/4), \\ 1, & \omega \in [3/4, 1]. \end{cases}

Problem 8.6

Let the probability space (Ω,F,P)(\Omega , \mathscr {F}, \mathbb {P}) be the line R\mathbb {R} with the σ\sigma-algebra of Borel sets and some probability measure. Describe the σ\sigma-algebra generated by the random variable:

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(a)

ξ(ω)=ω2\xi (\omega ) = \omega^2;

(b)

ξ(ω)=cos⁡ω\xi (\omega ) = \cos \omega.

Section 9Distribution Functions[52/77%]
Problem 9.1

Which of the following functions are distribution functions:

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(a)

  1x∈(0,1]\; \mathbb {1}_{x \in (0, 1]};

(b)

  1x∈(−∞,0]\; \mathbb {1}_{x \in (-\infty , 0]};

(c)

  1x∈(0,∞)\; \mathbb {1}_{x \in (0, \infty )};

(d)

  1x=0\; \mathbb {1}_{x = 0};

(e)

F(x)={0for x∈(−∞,0],1−1/nfor x∈(n−1,n], n∈N;F(x) = \begin{cases} 0 & \text{for } x \in (-\infty , 0], \\ 1 - 1/n & \text{for } x \in (n-1, n], \ n \in \mathbb {N}; \end{cases}

(f)

F(x)={0for x∈(−∞,0],1+(−1)n/nfor x∈(n−1,n], n∈N?F(x) = \begin{cases} 0 & \text{for } x \in (-\infty , 0], \\ 1 + (-1)^n/n & \text{for } x \in (n-1, n], \ n \in \mathbb {N}? \end{cases}

Problem 9.2

A discrete random variable ξ\xi has the distribution function

F(x)={0for x⩽2,3/10for 2<x⩽3,1/2for 3<x⩽4,1for x>4. F(x) = \begin{cases} 0 & \text{for } x \leqslant 2, \\ 3/10 & \text{for } 2 < x \leqslant 3, \\ 1/2 & \text{for } 3 < x \leqslant 4, \\ 1 & \text{for } x > 4. \end{cases}

Find the distribution series of the random variable:

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(a)

ξ\xi;

(b)

ξ2−2\xi^2 - 2.

Problem 9.3

A discrete random variable ξ\xi has the distribution series

xi−101P{ξ=xi}1/31/31/3 \begin{array}{c|ccc} x_i & -1 & 0 & 1 \\ \hline \mathbb {P}\left\{ \xi = x_i\right\} & 1/3 & 1/3 & 1/3 \end{array}

Construct the distribution series of the following random variables:

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(a)

∣ξ∣\left|\xi \right|;

(b)

2ξ+52\xi + 5;

(c)

ξ2+1\xi^2 + 1;

(d)

2ξ2^\xi;

(e)

ξ(ξ2−1)\xi (\xi^2 - 1);

(f)

min⁡(ξ,1)\min (\xi , 1).

Problem 9.4

A discrete random variable ξ\xi has the distribution series

xi−2−1012P{ξ=xi}1/101/53/103/101/10 \begin{array}{c|ccccc} x_i & -2 & -1 & 0 & 1 & 2 \\ \hline \mathbb {P}\left\{ \xi = x_i\right\} & 1/10 & 1/5 & 3/10 & 3/10 & 1/10 \end{array}

Construct the distribution series of the following random variables:

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(a)

3−ξ3 - \xi;

(b)

ξ2−1\xi^2 - 1;

(c)

∣ξ∣\left|\xi \right|;

(d)

max⁡(∣ξ∣,1)\max (\left|\xi \right|, 1);

(e)

min⁡(ξ,1)\min (\xi , 1);

(f)

1/(3−ξ)1/(3 - \xi ).

Problem 9.5

A discrete random variable ξ\xi has the distribution series

xi−2015P{ξ=xi}1/41/31/61/4 \begin{array}{c|cccc} x_i & -2 & 0 & 1 & 5 \\ \hline \mathbb {P}\left\{ \xi = x_i\right\} & 1/4 & 1/3 & 1/6 & 1/4 \end{array}

Construct the distribution series of the following random variables:

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(a)

1−2ξ1 - 2\xi;

(b)

ξ2+ξ−2\xi^2 + \xi - 2;

(c)

∣∣ξ∣−3∣\left|\left|\xi \right| - 3\right|;

(d)

2ξ2^\xi;

(e)

max⁡(ξ,1)\max (\xi , 1);

(f)

1/(3−ξ)1/(3 - \xi ).

Problem 9.6

To attract customers, the «Coca-Cola» company runs a promotion in which every tenth bottle of the drink the company produces is a winning bottle. Construct the distribution series of the number of winning bottles among three bottles purchased by a customer. Construct the distribution function of this random variable.

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Problem 9.7

Among 7 purchased theater tickets, 3 are for the orchestra section. 4 tickets are drawn at random. Construct the distribution series of the number of orchestra-section tickets among those chosen. Construct the distribution function of this random variable.

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Problem 9.8

Quality control inspects items, each of which, independently of the others, may turn out to be defective with probability pp. Find the distribution series of the number of good items found during inspection between two consecutive defective ones.

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Problem 9.9

Let the probability space (Ω,F,P)(\Omega , \mathscr {F}, \mathbb {P}) be the segment [0,1][0, 1] with the σ\sigma-algebra of Borel sets and Lebesgue measure. Find the distribution function of the random variable ξ\xi if:

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(a)

ξ(ω)=ω\xi (\omega ) = \omega;

(b)

ξ(ω)=ωα\xi (\omega ) = \omega^\alpha, α<0\alpha < 0;

(c)

ξ(ω)=ω2\xi (\omega ) = \omega^2;

(d)

ξ(ω)=sin⁡(πω)\xi (\omega ) = \sin (\pi \omega );

(e)

ξ(ω)={2ω,ω∈[0,1/2),2(1−ω),ω∈[1/2,1];\xi (\omega ) = \begin{cases} 2\omega , & \omega \in [0, 1/2), \\ 2(1-\omega ), & \omega \in [1/2, 1]; \end{cases}

(f)

ξ(ω)={ω,ω∈[0,1/3),−1,ω∈[1/3,2/3),−ω3,ω∈[2/3,1];\xi (\omega ) = \begin{cases} \omega , & \omega \in [0, 1/3), \\ -1, & \omega \in [1/3, 2/3), \\ -\omega^3, & \omega \in [2/3, 1]; \end{cases}

(g)

ξ(ω)={1/4,ω∈[0,1/4),1,ω∈[1/4,3/4),1/4,ω∈[3/4,1].\xi (\omega ) = \begin{cases} 1/4, & \omega \in [0, 1/4), \\ 1, & \omega \in [1/4, 3/4), \\ 1/4, & \omega \in [3/4, 1]. \end{cases}

Problem 9.10

Find the probability densities (where they exist) of the random variables defined in problem 9.9.

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(a)
(b)
(c)
(d)
(e)
(f)
(g)
Problem 9.11

Find the median of the distribution of the random variables defined in problem 9.9.

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(a)
(b)
(c)
(d)
(e)
(f)
(g)
Problem 9.12

A point ω=(ω1,ω2)\omega = (\omega_1, \omega_2) is chosen at random in the triangle with vertices at the points (0,0)(0, 0), (2,1)(2, 1), and (2,0)(2, 0). Find the distribution function and the density of the random variable ξ\xi if:

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(a)

ξ(ω)=ω1\xi (\omega ) = \omega_1;

(b)

ξ(ω)=ω2\xi (\omega ) = \omega_2.

Problem 9.13

An isosceles triangle is formed by a unit vector in the direction of the abscissa axis and a unit vector in a random direction (that is, with a direction uniformly distributed over all directions in the corresponding space). Find the distribution function of the length of the third side (the length of the difference of these two unit vectors) in:

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(a)

R2\mathbb {R}^{2};

(b)

R3\mathbb {R}^{3}.

Problem 9.14

A line is drawn from the point (0,a)(0, a) in the plane at an angle ϕ\phi to the ordinate axis. Find the distribution function of the point where this line intersects the abscissa axis, if the angle ϕ\phi is uniformly distributed on the interval:

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(a)

(0,π/2)(0, \pi /2);

(b)

(−π/2,π/2)(-\pi /2, \pi /2).

Problem 9.15

A point is thrown at random onto a circle of radius RR centered at the origin. Find:

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(a)

the density of the abscissa of the point hit;

(b)

the density of the length of the chord joining the point hit with the point (−R,0)(-R, 0).

Problem 9.16

A point is thrown at random onto the segment of the ordinate axis between the points (0,0)(0, 0) and (0,R)(0, R). Through the point hit, a chord of the circle x2+y2=R2x^2 + y^2 = R^2 is drawn perpendicular to the ordinate axis. Find the distribution of the length of this chord.

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Problem 9.17

The diameter of a circle is measured approximately. Assuming its value is uniformly distributed on the segment [a,b][a, b], find the distribution of the area of the circle.

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Problem 9.18

Let p(x)p(x) be a probability density. Find the value of the constant cc appearing in the definition of p(x)p(x):

p(x)={ce−αxfor x>0,0for x⩽0, p(x) = \begin{cases} ce^{-\alpha x} & \text{for } x > 0, \\ 0 & \text{for } x \leqslant 0, \end{cases}

where α>0\alpha > 0.

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Problem 9.19

Let p(x)p(x) be a probability density. Find the value of the constant cc appearing in the definition of p(x)p(x):

p(x)=c1+(x−α)2,α∈R. p(x) = \frac{c}{1 + (x - \alpha )^2}, \quad \alpha \in \mathbb {R}.
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Problem 9.20

Let p(x)p(x) be a probability density. Find the value of the constant cc appearing in the definition of p(x)p(x):

p(x)={cxβe−αxfor x>0,0for x⩽0,α>0, β>0. p(x) = \begin{cases} cx^\beta e^{-\alpha x} & \text{for } x > 0, \\ 0 & \text{for } x \leqslant 0, \end{cases} \quad \alpha > 0, \ \beta > 0.
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Problem 9.21

Let p(x)p(x) be a probability density. Find the value of the constant cc appearing in the definition of p(x)p(x):

p(x)=ce−x+ex. p(x) = \frac{c}{e^{-x} + e^x}.
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Problem 9.22

The probability density of the random variable ξ\xi has the form

p(x)={0for x<1,1/3for 1⩽x⩽c,0for x>c. p(x) = \begin{cases} 0 & \text{for } x < 1, \\ 1/3 & \text{for } 1 \leqslant x \leqslant c, \\ 0 & \text{for } x > c. \end{cases}

Find:

?
(a)

the value of the constant cc;

(b)

the distribution function of the random variable ξ\xi;

(c)

the probability P{ξ⩾2}\mathbb {P}\left\{ \xi \geqslant 2\right\}.

Problem 9.23

The probability density of the random variable ξ\xi has the form

p(x)={1/5for −1⩽x<2,cfor 2⩽x⩽3,0otherwise. p(x) = \begin{cases} 1/5 & \text{for } -1 \leqslant x < 2, \\ c & \text{for } 2 \leqslant x \leqslant 3, \\ 0 & \text{otherwise.} \end{cases}

Find:

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(a)

the value of the constant cc;

(b)

the distribution function of the random variable ξ\xi;

(c)

the probability P{0⩽ξ⩽2,5}\mathbb {P}\left\{ 0 \leqslant \xi \leqslant 2{,}5\right\}.

Problem 9.24

Can a constant cc be chosen so that the function cx−4cx^{-4} is a probability density on the set:

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(a)

[1,∞)[1, \infty );

(b)

[0,∞)[0, \infty );

(c)

[−2,−1][-2, -1];

(d)

[−3,0)[-3, 0)?

Problem 9.25

Plot the density graph of:

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(a)

the standard normal distribution;

(b)

the normal distribution with parameters aa and σ2\sigma^2;

(c)

the uniform distribution on the segment [a,b][a, b];

(d)

the exponential distribution with parameter α\alpha;

(e)

the Cauchy distribution with parameters 0 and 1.

Problem 9.26

What happens to the graph of the distribution function of the random variable ξ\xi if:

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(a)

the number 3 is added to ξ\xi;

(b)

ξ\xi is multiplied by the number 3?

Problem 9.27

What happens to the density graph of an absolutely continuous distribution of the random variable ξ\xi if:

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(a)

the number 3 is added to ξ\xi;

(b)

ξ\xi is multiplied by the number 3?

Problem 9.28

Let the random variable ξ\xi have a normal distribution with parameters aa and σ2\sigma^2. Prove that the random variable η=ξ−aσ\eta = \dfrac {\xi - a}{\sigma } has the standard normal distribution.

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Problem 9.29

Let the random variable ξ\xi have a normal distribution. It is known that P{ξ>2}=0,5\mathbb {P}\left\{ \xi > 2\right\} = 0{,}5 and P{ξ<3}=0,975\mathbb {P}\left\{ \xi < 3\right\} = 0{,}975. Find the probability density of ξ\xi and P{1<ξ<3}\mathbb {P}\left\{ 1 < \xi < 3\right\}.

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Problem 9.30

Find the expectation and variance of the normally distributed random variable ξ\xi, given that P{ξ<0,44}=0,1\mathbb {P}\left\{ \xi < 0{,}44\right\} = 0{,}1 and P{ξ⩾3,88}=0,33\mathbb {P}\left\{ \xi \geqslant 3{,}88\right\} = 0{,}33.

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Problem 9.31

Let the random variable ξ\xi be uniformly distributed on the segment [0,1][0, 1]. Find the probability densities of the following random variables:

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(a)

−ln⁡ξ-\ln \xi;

(b)

2ξ+12\xi + 1;

(c)

ξ−1/ξ\xi - 1/\xi;

(d)

−ln⁡(1−ξ)-\ln (1 - \xi );

(e)

ξ2\xi^2;

(f)

eξ−1e^\xi - 1.

Problem 9.32

Let the random variable ξ\xi have an exponential distribution with parameter α\alpha. Find the probability densities of the following random variables:

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(a)

ξ\sqrt{\xi };

(b)

ξ2\xi^2;

(c)

2ξ2^\xi;

(d)

ln⁡(αξ)\ln (\alpha \xi );

(e)

e−αξe^{-\alpha \xi };

(f)

min⁡(ξ,ξ2)\min (\xi , \xi^2).

Problem 9.33

Let the random variable ξ\xi have the standard Cauchy distribution (a=0a = 0, σ2=1\sigma^2 = 1). Find the probability densities of the following random variables:

?
(a)

2ξ+12\xi + 1;

(b)

1/ξ1/\xi;

(c)

1/(1+ξ2)1/(1 + \xi^2);

(d)

ξ2/(1+ξ2)\xi^2/(1 + \xi^2).

Problem 9.34

Let the random variable ξ\xi have the standard normal distribution. Find the probability density of the random variable η=eξ\eta = e^\xi (this density is called the log-normal density).

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Problem 9.35

Let the random variable ξ\xi be uniformly distributed on the segment [0,2π][0, 2\pi ]. Find the probability density of the tangent of ξ\xi.

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Problem 9.36

Let the random variable ξ\xi have probability density p(x)p(x). Find the probability densities of the following quantities:

?
(a)

aξ+ba\xi + b, a,b∈Ra, b \in \mathbb {R}, a≠0a \neq 0;

(b)

ξ−1\xi^{-1};

(c)

ξ2\xi^2;

(d)

min⁡(ξ,ξ2)\min (\xi , \xi^2);

(e)

cos⁡ξ\cos \xi;

(f)

the fractional part of ξ\xi;

(g)

∣ξ−1∣\left|\xi - 1\right|;

(h)

max⁡(ξ,ξ2)\max (\xi , \xi^2).

Problem 9.37

Prove that for any random variable ξ\xi with continuous distribution function FF, the equality

P{F(ξ)<x}=x,x∈[0,1] \mathbb {P}\left\{ F(\xi ) < x\right\} = x, \quad x \in [0, 1]

holds.

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Problem 9.38

What can be said about the distribution of the random variable F(ξ)F(\xi ) in the previous problem, if the distribution function FF is not continuous?

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Problem 9.39

Let the random variable ξ\xi be uniformly distributed on the segment [0,1][0, 1]. Prove that for any distribution function FF there exists a measurable function ff such that the random variable η=f(ξ)\eta = f(\xi ) has distribution function FF.

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Problem 9.40

Let the random variable ξ\xi have continuous distribution function FF. Find the distribution function of the random variable η=−ln⁡F(ξ)\eta = -\ln F(\xi ).

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Problem 9.41

Give an example of a random variable ξ\xi with an absolutely continuous distribution and a continuous function gg such that g(ξ)g(\xi ) has a non-degenerate discrete distribution.

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Problem 9.42

The random variable ξ\xi has distribution function FF. Find the distribution function of the random variable η=∣ξ∣s\eta = \left|\xi \right|^s, where s>0s > 0.

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Problem 9.43

Let the distribution function FF of the random variable ξ\xi be continuous at zero. Find the distribution of the random variable η=ξ/∣ξ∣\eta = \xi /\left|\xi \right|.

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Problem 9.44

The random variable ξ\xi has distribution function FF. Find the distribution functions of the random variables:

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(a)

max⁡(0,ξ)\max (0, \xi );

(b)

min⁡(0,ξ)\min (0, \xi );

(c)

(ξ+∣ξ∣)/2(\xi + \left|\xi \right|)/2;

(d)

2−3ξ2 - 3\xi.

Problem 9.45

Prove that if the random variable ξ\xi has an absolutely continuous distribution, then ∣ξ∣\left|\xi \right| also has an absolutely continuous distribution. Is the converse statement true?

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Problem 9.46

Prove that any distribution function FF has the following properties:

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(a)

lim⁡x→∞x∫x∞1z dF(z)=0\displaystyle \lim_{x \to \infty } x \int_x^\infty \frac{1}{z}\, dF(z) = 0;

(b)

lim⁡x→+0x∫x∞1z dF(z)=0\displaystyle \lim_{x \to +0} x \int_x^\infty \frac{1}{z}\, dF(z) = 0.

Problem 9.47

Prove that for any continuous distribution function FF the following equalities hold:

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(a)

∫−∞∞F(x) dF(x)=12\displaystyle \int_{-\infty }^\infty F(x)\, dF(x) = \frac{1}{2};

(b)

∫−∞∞F2(x) dF(x)=13\displaystyle \int_{-\infty }^\infty F^2(x)\, dF(x) = \frac{1}{3}.

Problem 9.48

Prove that if F(x)F(x) is a distribution function, then for any h>0h > 0 the functions:

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(a)

G(x)≡1h∫xx+hF(y) dyG(x) \equiv \dfrac {1}{h}\displaystyle \int_x^{x+h} F(y)\, dy;

(b)

H(x)≡12h∫x−hx+hF(y) dyH(x) \equiv \dfrac {1}{2h}\displaystyle \int_{x-h}^{x+h} F(y)\, dy

are also distribution functions.

Problem 9.49

Prove that the set of discontinuity points of a distribution function is at most countable.

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Problem 9.50

Prove that any distribution FF can be represented in the form F=αFd(x)+(1−α)Fa(x)F = \alpha F_d(x) + (1 - \alpha ) F_a(x), where α∈[0,1]\alpha \in [0, 1], Fd(x)F_d(x) is some discrete distribution, and Fa(x)F_a(x) is continuous.

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Problem 9.51

Prove that a distribution function is uniquely determined by its values at an at-most-countable set of points.

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Problem 9.52

What is the cardinality of the set of all distribution functions?

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Section 10Joint Distribution. Independence[39/74%]
Problem 10.1

The two-dimensional distribution of a pair of integer-valued random variables ξ\xi and η\eta is given by the table

η\ξ−101−11/81/127/2415/241/61/8 \begin{array}{c|ccc} {}_\eta \backslash ^\xi & -1 & 0 & 1 \\ \hline -1 & 1/8 & 1/12 & 7/24 \\ 1 & 5/24 & 1/6 & 1/8 \end{array}

where the entry at the intersection of column ξ=i\xi = i and row η=j\eta = j is the probability P{ξ=i,η=j}\mathbb {P}\left\{ \xi = i, \eta = j\right\}. Determine whether the random variables ξ\xi and η\eta are dependent. Find:

?
(a)

the one-dimensional distributions P{ξ=i}\mathbb {P}\left\{ \xi = i\right\} and P{η=j}\mathbb {P}\left\{ \eta = j\right\};

(b)

the conditional probabilities P{η=1∣ξ=0}\mathbb {P}\left\{ \eta = 1 \mid \xi = 0\right\} and P{ξ=1∣η=1}\mathbb {P}\left\{ \xi = 1 \mid \eta = 1\right\};

(c)

the joint distribution of the quantities ξ+η\xi + \eta and ξη\xi \eta;

(d)

the one-dimensional distributions of the quantities ξ+η\xi + \eta and ξη\xi \eta;

(e)

the joint distribution of the quantities max⁡(ξ,η)\max (\xi , \eta ) and min⁡(ξ,η)\min (\xi , \eta );

(f)

the joint distribution of the quantities ∣ξ+2η∣\left|\xi + 2\eta \right| and ∣ξ−η∣\left|\xi - \eta \right|.

Problem 10.2

Solve problem 10.1 for the table

η\ξ−20111/61/61/621/61/61/6 \begin{array}{c|ccc} {}_\eta \backslash ^\xi & -2 & 0 & 1 \\ \hline 1 & 1/6 & 1/6 & 1/6 \\ 2 & 1/6 & 1/6 & 1/6 \end{array}
?
(a)
(b)
(c)
(d)
(e)
Problem 10.3

Solve problem 10.1 for the table

η\ξ−101−21/81/127/2401/121/121/1611/81/121/16 \begin{array}{c|ccc} {}_\eta \backslash ^\xi & -1 & 0 & 1 \\ \hline -2 & 1/8 & 1/12 & 7/24 \\ 0 & 1/12 & 1/12 & 1/16 \\ 1 & 1/8 & 1/12 & 1/16 \end{array}
?
(a)
(b)
(c)
(d)
(e)
(f)
Problem 10.4

A number ξ\xi is chosen at random among the numbers 1, 2, and 3; a number η\eta is chosen at random among ξ,ξ+1,…,ξ+3\xi , \xi +1, \ldots , \xi +3. Find:

?
(a)

the joint distribution of the numbers ξ\xi and η\eta;

(b)

the one-dimensional distribution of ξ\xi;

(c)

the one-dimensional distribution of η\eta.

Problem 10.5

Let the random variables ξ\xi and η\eta be independent and identically distributed, with P{ξ=1}=p\mathbb {P}\left\{ \xi = 1\right\} = p and P{ξ=0}=1−p\mathbb {P}\left\{ \xi = 0\right\} = 1 - p. Introduce a new random variable ζ\zeta, setting

ζ={0,if ξ+η is even,1,if ξ+η is odd. \zeta = \begin{cases} 0, & \text{if } \xi + \eta \text{ is even}, \\ 1, & \text{if } \xi + \eta \text{ is odd.} \end{cases}

For which values of pp are the random variables ξ\xi and ζ\zeta independent?

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Problem 10.6

Let ξ1\xi_1 and ξ2\xi_2 be independent random variables taking integer values. Under what conditions are the events {ξ1 is even}\left\{ \xi_1 \text{ is even}\right\} and {ξ1+ξ2 is even}\left\{ \xi_1 + \xi_2 \text{ is even}\right\} independent?

?
Problem 10.7

Let the independent random variables ξ\xi, η\eta, and ζ\zeta have the same geometric distribution with parameter pp. Find:

?
(a)

P{ξ=η}\mathbb {P}\left\{ \xi = \eta \right\};

(b)

P{ξ⩾2η}\mathbb {P}\left\{ \xi \geqslant 2\eta \right\};

(c)

P{ξ+η⩽ζ}\mathbb {P}\left\{ \xi + \eta \leqslant \zeta \right\}.

Problem 10.8

A point ω=(ω1,ω2)\omega = (\omega_1, \omega_2) is chosen at random in the square [0,1]2[0, 1]^2. Find the distribution function and density of the random variable ξ+η\xi + \eta if:

?
(a)

ξ(ω)=ω1+ω2\xi (\omega ) = \omega_1 + \omega_2, η(ω)=ω1−ω2\eta (\omega ) = \omega_1 - \omega_2;

(b)

ξ(ω)=ω1\xi (\omega ) = \omega_1, η(ω)=ω2\eta (\omega ) = \omega_2;

(c)

ξ(ω)=1\xi (\omega ) = 1 if ω1=ω2\omega_1 = \omega_2 and ξ(ω)=0\xi (\omega ) = 0 if ω1≠ω2\omega_1 \neq \omega_2; η(ω)=ω1ω2\eta (\omega ) = \omega_1\omega_2.

Problem 10.9

Consider in the plane a region Ω\Omega of area 1/21/2 that is the union of the quadrilateral with vertices (0,0)(0,0), (1,1)(1,1), (0,1/2)(0,1/2), (1/2,1)(1/2,1) and the triangle with vertices (1/2,0)(1/2,0), (1,0)(1,0), (1,1/2)(1,1/2). A random point (ξ,η)(\xi , \eta ) is uniformly distributed in Ω\Omega. Prove that:

?
(a)

ξ\xi and η\eta are uniformly distributed on [0,1][0, 1];

(b)

the sum ξ+η\xi + \eta has the same density as it would if ξ\xi and η\eta were independent.

Problem 10.10

For a given tram line, the function F(a,b)F(a, b) is known, equal to the probability that a passenger riding this line boarded at a point x<ax < a and rides to a point y⩽by \leqslant b. Find the probability that a passenger:

?
(a)

boarded the tram before point zz;

(b)

got off before point zz;

(c)

passes through point zz.

Problem 10.11

Let F(x,y)F(x, y) denote a two-dimensional distribution function. Set G(x,y)=0G(x, y) = 0 for x<0x < 0 and y<0y < 0; G(x,y)=F(x,y)G(x, y) = F(x, y) at all other points. Prove that GG is monotone in each variable, but GG need not be a distribution function.

?
Problem 10.12

Let ξ\xi and η\eta be random variables with distribution functions FF and GG respectively. Prove that if P{ξ⩽η}=1\mathbb {P}\left\{ \xi \leqslant \eta \right\} = 1, then F(x)⩾G(x)F(x) \geqslant G(x) for all xx.

?
Problem 10.13

Two points are thrown independently at random onto the segment [0,a][0, a]. Find the distribution function of the distance between them.

?
Problem 10.14

Let p(x,y)p(x, y) be the joint density of the random variables ξ\xi and η\eta. Find:

?
(a)

P{ξ>η}\mathbb {P}\left\{ \xi > \eta \right\};

(b)

P{ξ>∣η∣}\mathbb {P}\left\{ \xi > \left|\eta \right|\right\};

(c)

P{max⁡(ξ,η)<a}\mathbb {P}\left\{ \max (\xi , \eta ) < a\right\};

(d)

P{∣ξ∣>η}\mathbb {P}\left\{ \left|\xi \right| > \eta \right\};

(e)

P{ξ−η>1}\mathbb {P}\left\{ \xi - \eta > 1\right\};

(f)

P{min⁡(ξ,η)<a}\mathbb {P}\left\{ \min (\xi , \eta ) < a\right\}.

Problem 10.15

Let the random variables ξ\xi and η\eta be independent and have the same continuous distribution function. Find:

?
(a)

P{ξ=η}\mathbb {P}\left\{ \xi = \eta \right\};

(b)

P{ξ<η}\mathbb {P}\left\{ \xi < \eta \right\}.

Problem 10.16

Let the independent random variables ξ\xi and η\eta have exponential distributions with parameters α\alpha and β\beta respectively. Find P{ξ<η}\mathbb {P}\left\{ \xi < \eta \right\}.

?
Problem 10.17

The joint distribution of the random variables ξ\xi and η\eta has density

p(x,y)={c(x+y)for 0⩽x⩽1, 0⩽y⩽1,0otherwise. p(x, y) = \begin{cases} c(x+y) & \text{for } 0 \leqslant x \leqslant 1,\ 0 \leqslant y \leqslant 1, \\ 0 & \text{otherwise.} \end{cases}

Find:

?
(a)

the constant cc;

(b)

the density of ξ\xi;

(c)

the density of η\eta;

(d)

the density of max⁡(ξ,η)\max (\xi , \eta );

(e)

the density of min⁡(ξ,η)\min (\xi , \eta ).

Problem 10.18

The random variable ξ\xi has probability density p(x)p(x). Find the joint distribution function of the random vector (ξ,ξ2)(\xi , \xi^2).

?
Problem 10.19

The joint distribution of the random variables ξ\xi and η\eta has density

p(x,y)={1,if x,y⩾0 and x+2y⩽2,0otherwise. p(x, y) = \begin{cases} 1, & \text{if } x, y \geqslant 0 \text{ and } x + 2y \leqslant 2, \\ 0 & \text{otherwise.} \end{cases}

Find the density of the random variable ξ\xi.

?
Problem 10.20

The joint distribution of the random variables ξ\xi and η\eta has density

p(x,y)=c1+x2+x2y2+y2. p(x, y) = \frac{c}{1 + x^2 + x^2y^2 + y^2}.

Determine whether the random variables ξ\xi and η\eta are dependent or not. Find:

?
(a)

the constant cc;

(b)

the density of ξ\xi;

(c)

the density of η\eta;

(d)

the probability P{∣ξ∣⩽1,∣η∣⩽1}\mathbb {P}\left\{ \left|\xi \right| \leqslant 1, \left|\eta \right| \leqslant 1\right\}.

Problem 10.21

A random point (ξ,η)(\xi , \eta ) is uniformly distributed inside the square {(x,y):∣x∣+∣y∣⩽1}\left\{ (x, y) : \left|x\right| + \left|y\right| \leqslant 1\right\}. Determine whether the random variables ξ\xi and η\eta are dependent or not. Find:

?
(a)

the density of ξ\xi;

(b)

the density of η\eta;

(c)

the probability P{ξ>η}\mathbb {P}\left\{ \xi > \eta \right\}.

Problem 10.22

The joint distribution of the random variables ξ\xi and η\eta has density

p(x,y)={2π(x2+y2)3for x2+y2⩾1,0otherwise. p(x, y) = \begin{cases} \dfrac {2}{\pi (x^2+y^2)^3} & \text{for } x^2 + y^2 \geqslant 1, \\ 0 & \text{otherwise.} \end{cases}

Find the density of ξ2+η2\sqrt{\xi^2 + \eta^2}.

?
Problem 10.23

A random point (ξ,η)(\xi , \eta ) has a normal distribution on the plane with density

pξη(x,y)=12π2e−(2(x−1)2+(y+1)2)/4. p_{\xi \eta }(x, y) = \frac{1}{2\pi \sqrt{2}} e^{-(2(x-1)^2 + (y+1)^2)/4}.

Find the probability that the random point (ξ,η)(\xi , \eta ) lands in the region bounded by the ellipse

(x−1)2+(y+1)22=1. \frac{(x-1)^2 + (y+1)^2}{2} = 1.
?
Problem 10.24

The random variables ξ\xi and η\eta are independent and have the standard normal distribution. Find the probability that the random point (ξ,η)(\xi , \eta ) lands

?
(a)

in the annulus {(x,y):2⩽x2+y2⩽3}\left\{ (x, y) : 2 \leqslant \sqrt{x^2+y^2} \leqslant 3\right\};

(b)

in the square {(x,y):∣x∣+∣y∣⩽2}\left\{ (x, y) : \left|x\right| + \left|y\right| \leqslant \sqrt{2}\right\}.

Problem 10.25

A random point (ξ,η)(\xi , \eta ) has a uniform distribution in the disk {(x,y):x2+y2<1}\left\{ (x, y) : x^2 + y^2 < 1\right\}.

?
(a)

Find the probability P{∣ξ∣<1/2,∣η∣<1/2}\mathbb {P}\left\{ \left|\xi \right| < 1/\sqrt{2}, \left|\eta \right| < 1/\sqrt{2}\right\}.

(b)

Are the random variables ξ\xi and η\eta independent?

Problem 10.26

The random vector (ξ,η,ζ)(\xi , \eta , \zeta ) has probability density

p(x,y,z)={6(1+x+y+z)4for x>0, y>0, z>0,0otherwise. p(x, y, z) = \begin{cases} \dfrac {6}{(1+x+y+z)^4} & \text{for } x > 0,\ y > 0,\ z > 0, \\ 0 & \text{otherwise.} \end{cases}

Find the distribution of the quantity ξ+η+ζ\xi + \eta + \zeta.

?
Problem 10.27

Two points are chosen randomly and independently on the segment [0,1][0, 1]; a third point is chosen at random between the first two. Prove that the distribution of the coordinate of the third point is absolutely continuous. Find its density.

?
Problem 10.28

Show that if ξ1,ξ2,…\xi_1, \xi_2, \ldots are independent random variables, then both random variables lim inf⁡n→∞ξn\liminf_{n \to \infty } \xi_n and lim sup⁡n→∞ξn\limsup_{n \to \infty } \xi_n are degenerate.

?
Problem 10.29

Let the random variables ξ\xi and η\eta be such that P{ξ>0}=P{η>0}=3/4\mathbb {P}\left\{ \xi > 0\right\} = \mathbb {P}\left\{ \eta > 0\right\} = 3/4 and P{ξ+η>0}=1/2\mathbb {P}\left\{ \xi + \eta > 0\right\} = 1/2. Prove that ξ\xi and η\eta are dependent.

?
Problem 10.30

Let ξ\xi have the standard normal distribution, and let η=1\eta = 1 if ξ>0\xi > 0 and η=0\eta = 0 if ξ⩽0\xi \leqslant 0. Prove that the random variables ∣ξ∣\left|\xi \right| and η\eta are independent.

?
Problem 10.31

Let the independent random variables ξ\xi and η\eta have distribution functions F(x)F(x) and G(x)G(x) respectively. Find the distribution functions of the following quantities:

?
(a)

max⁡(ξ,η)\max (\xi , \eta );

(b)

min⁡(ξ,η)\min (\xi , \eta );

(c)

max⁡(3ξ,η)\max (3\xi , \eta );

(d)

min⁡(ξ,η3)\min (\xi , \eta^3).

Problem 10.32

Three customers approach a telephone exchange with two booths: the first and second customers occupy booths No.1 and 2 respectively, and the third customer is left waiting. Assuming the call durations τ1\tau_1, τ2\tau_2, and τ3\tau_3 of the customers are independent and exponentially distributed with parameter α\alpha, find:

?
(a)

the probability that the third customer will get booth No.1;

(b)

the density of the waiting time of the third customer;

(c)

the probability that the third customer finishes talking before the first or the second customer (i.e., before at least one of them).

Problem 10.33

When is a random variable ξ\xi independent of itself?

?
Problem 10.34

Under what conditions on ξ\xi are the random variables ξ\xi and sin⁡ξ\sin \xi independent?

?
Problem 10.35

Prove that if the random variables ξ\xi and f(ξ)f(\xi ) (ff a Borel function) are independent, then f(ξ)f(\xi ) equals some constant with probability 1.

?
Problem 10.36

Prove that if the random variables ξ\xi and η\eta, and also ξ\xi and ξ−η\xi - \eta, are independent, then ξ\xi equals some constant with probability 1.

?
Problem 10.37

Let ξ\xi and η\eta be random variables. Must they be independent if the random variables ξ2\xi^2 and η2\eta^2 are independent?

?
Problem 10.38

Let the independent random variables ξ\xi, η\eta, and ζ\zeta take only the values 0,1,2,…0, 1, 2, \ldots, each with positive probability. Prove that there is no integer n⩾2n \geqslant 2 for which the equality ξn+ηn=ζn\xi^n + \eta^n = \zeta^n holds with probability 1.

?
Problem 10.39

Let ξ\xi and η\eta be independent random variables, with ξ+η\xi + \eta taking the values 0, 1, and 2 each with probability 1/31/3. Prove that one of the variables ξ\xi or η\eta has a degenerate distribution.

?
Section 11Transformations of Independent Random Variables[45/78%]
Problem 11.1

Let the discrete random variables ξ\xi and η\eta be independent and have Poisson distributions with parameters λ\lambda and μ\mu respectively. Find the distribution of the sum ξ+η\xi + \eta.

?
Problem 11.2

Let the discrete random variables ξ1,…,ξn\xi_1, \ldots , \xi_n be independent and have Poisson distributions with parameters λ1,…,λn\lambda_1, \ldots , \lambda_n respectively. Find the distribution of their sum ξ1+…+ξn\xi_1 + \ldots + \xi_n.

?
Problem 11.3

Let the random variables ξ\xi and η\eta be independent, with ξ\xi having a binomial distribution with parameters pp and nn, and η\eta a binomial distribution with parameters pp and mm. Find the distribution of the sum ξ+η\xi + \eta.

?
Problem 11.4

Let ξ1,…,ξn\xi_1, \ldots , \xi_n be independent random variables, each taking the values 0 and 1 with probability 1/21/2. Find the distribution of the product η=∏i=1nξi\eta = \prod_{i=1}^n \xi_i.

?
Problem 11.5

Let the independent random variables ξ\xi and η\eta have the same geometric distribution with parameter pp. Set U=min⁡(ξ,η)U = \min (\xi , \eta ) and V=ξ−ηV = \xi - \eta. Prove that the random variables UU and VV are independent.

?
Problem 11.6

Let ξ\xi and η\eta be independent random variables, with ξ\xi uniformly distributed on the segment [0,a][0, a] and η\eta uniformly distributed on the segment [0,b][0, b]. Find the distribution of the sum ξ+η\xi + \eta if:

?
(a)

a=b=1a = b = 1;

(b)

b>ab > a.

Problem 11.7

Let the random variables ξ\xi and η\eta be independent, each uniformly distributed on the segment [0,2][0, 2]. Find the distribution functions and densities of the following quantities:

?
(a)

∣ξ−η∣\left|\xi - \eta \right|;

(b)

min⁡(ξ,η3)\min (\xi , \eta^3);

(c)

ξ−3η\xi - 3\eta;

(d)

max⁡(2ξ,η2)\max (2\xi , \eta^2);

(e)

max⁡(3ξ,η3)\max (3\xi , \eta^3).

Problem 11.8

Solve problem 11.7 assuming that the random variables ξ\xi and η\eta have an exponential distribution with parameter α\alpha.

?
(a)
(b)
(c)
(d)
(e)
Problem 11.9

nn points are thrown at random and independently of one another onto the segment [0,1][0, 1]. Find:

?
(a)

the density of the coordinate of the rightmost point;

(b)

the density of the coordinate of the leftmost point;

(c)

the density of the coordinate of the kk-th point from the left;

(d)

the joint density of the coordinates of the leftmost and rightmost points;

(e)

the joint density of the kk-th and mm-th points from the left (k<mk < m);

(f)

the density of the distance between the leftmost and rightmost points.

Problem 11.10

Let the random variables ξ1,…,ξn\xi_1, \ldots , \xi_n be independent, each with a symmetric distribution. Prove that the sum ξ1+…+ξn\xi_1 + \ldots + \xi_n also has a symmetric distribution.

?
Problem 11.11

The quantity Qξ(y)=sup⁡xP{x⩽ξ⩽x+y}Q_\xi (y) = \sup_x \mathbb {P}\left\{ x \leqslant \xi \leqslant x+y\right\} is called the concentration function of the random variable ξ\xi. Prove that for any random variable η\eta independent of ξ\xi, the concentration function of the sum ξ+η\xi + \eta satisfies the inequality

Qξ+η(y)⩽Qξ(y). Q_{\xi +\eta }(y) \leqslant Q_\xi (y).
?
Problem 11.12

Let the random variables ξ\xi and η\eta be independent, with ξ\xi having a normal distribution with parameters aa and σ2\sigma^2, and η\eta a normal distribution with parameters bb and θ2\theta^2. Find the distribution of the sum ξ+η\xi + \eta.

?
Problem 11.13

Let the random variables ξ1,…,ξn\xi_1, \ldots , \xi_n be independent, with ξi\xi_i having a normal distribution with parameters aia_i and σi2\sigma_i^2. Find the distribution of the sum ξ1+…+ξn\xi_1 + \ldots + \xi_n.

?
Problem 11.14

Let the independent random variables ξ\xi and η\eta have the standard normal distribution. Prove that the random variables ξ+η\xi + \eta and ξ−η\xi - \eta are independent.

?
Problem 11.15

The joint distribution of the random variables ξ\xi and η\eta has density

p(x,y)=12πe−(5x2−6xy+2y2)/2. p(x, y) = \frac{1}{2\pi } e^{-(5x^2 - 6xy + 2y^2)/2}.

Find the densities of:

?
(a)

ξ\xi;

(b)

η\eta;

(c)

ξ+η\xi + \eta;

(d)

the vector (ξ+η,ξ−η)(\xi + \eta , \xi - \eta ).

Problem 11.16

Let the random variables ξ1,…,ξn\xi_1, \ldots , \xi_n be independent and have the standard normal distribution. Find the joint density of the quantities η=∑k=1nξk\eta = \sum_{k=1}^n \xi_k and ζ=∑k=1mξk\zeta = \sum_{k=1}^m \xi_k for m<nm < n.

?
Problem 11.17

Let the independent random variables ξ\xi and η\eta have the standard normal distribution. Find the distribution of the random variable ξ2+η2\xi^2 + \eta^2.

?
Problem 11.18

Let the random variables ξ\xi and η\eta be independent and have the standard normal distribution. Prove that the quantities ξ2+η2\xi^2 + \eta^2 and ξ/η\xi /\eta are also independent.

?
Problem 11.19

Let ξ\xi and η\eta be independent random variables, with ξ\xi having an exponential distribution with parameter α\alpha and η\eta an exponential distribution with parameter β\beta. Find the distribution of the sum ξ+η\xi + \eta if:

?
(a)

α=β\alpha = \beta;

(b)

α<β\alpha < \beta.

Problem 11.20

Let ξ1,…,ξn\xi_1, \ldots , \xi_n be independent random variables, each having an exponential distribution with parameter α\alpha. Find the distribution of the sum ξ1+…+ξn\xi_1 + \ldots + \xi_n.

?
Problem 11.21

Let the random variables ξ\xi and η\eta be independent and have the Laplace distribution with parameter α\alpha. Find the density of the sum ξ+η\xi + \eta.

?
Problem 11.22

Let the random variables ξ\xi and η\eta be independent and have Cauchy distributions with the same scale parameter σ\sigma and location parameters aa and bb respectively. Find the density of the sum ξ+η\xi + \eta.

?
Problem 11.23

Let the random variables ξ1,…,ξn\xi_1, \ldots , \xi_n be independent and have Cauchy distributions with the same scale parameter σ\sigma and location parameters a1,…,ana_1, \ldots , a_n respectively. Find the density of the sum ξ1+…+ξn\xi_1 + \ldots + \xi_n.

?
Problem 11.24

Let the random variables ξ\xi and η\eta be independent, with ξ\xi having a gamma distribution with parameters α\alpha and β1\beta_1, and η\eta a gamma distribution with parameters α\alpha and β2\beta_2. Find the distribution of the sum ξ+η\xi + \eta.

?
Problem 11.25

Let the random variables ξ1,…,ξn\xi_1, \ldots , \xi_n be independent, with ξi\xi_i having a gamma distribution with parameters α\alpha and βi\beta_i. Find the distribution of the sum ξ1+…+ξn\xi_1 + \ldots + \xi_n.

?
Problem 11.26

Let a random variable have a uniform distribution on some segment. Determine whether it can be represented as the sum of two independent random variables with non-degenerate distributions.

?
Problem 11.27

Let ξ\xi and η\eta be independent random variables, with ξ\xi having an exponential distribution with parameter α\alpha and η\eta an exponential distribution with parameter β\beta. Find the distribution of the difference ξ−η\xi - \eta if:

?
(a)

α=β\alpha = \beta;

(b)

α<β\alpha < \beta.

Problem 11.28

Let ξ\xi, η\eta, and ζ\zeta be independent random variables having the same exponential distribution with parameter α\alpha. Find the joint density of the random vector (η−ξ,ζ−ξ)(\eta - \xi , \zeta - \xi ).

?
Problem 11.29

Find the density of the ratio ξ/η\xi /\eta, if the numerator and denominator are independent and each has:

?
(a)

an exponential distribution with parameter α\alpha;

(b)

a uniform distribution on the segment [0,1][0, 1];

(c)

the standard normal distribution.

Problem 11.30

The joint distribution of the random variables ξ\xi and η\eta has density p(x,y)=e−x−yp(x, y) = e^{-x-y}, x,y⩾0x, y \geqslant 0. Find:

?
(a)

the joint density of the random variables ξ+η\xi + \eta and ξ/η\xi /\eta;

(b)

prove that the quantities ξ+η\xi + \eta and ξ/η\xi /\eta are independent.

Problem 11.31

Find the density of the product ξη\xi \eta, if the factors are independent and each has a uniform distribution on the segment [0,1][0, 1].

?
Problem 11.32

A particle of unit mass splits at random into two fragments of masses ξ\xi and 1−ξ1-\xi. The density ff of the random variable ξ\xi is concentrated on [0,1][0, 1] and, by symmetry, f(x)=f(1−x)f(x) = f(1-x). Denote the mass of the smaller fragment by ξ1\xi_1 and of the larger by ξ2\xi_2. Suppose these two fragments, independently of each other, split in the same way and yield four fragments of masses ξ11,ξ12,ξ21\xi_{11}, \xi_{12}, \xi_{21}, and ξ22\xi_{22} (that is, ξ11\xi_{11} and ξ12\xi_{12} are the masses of the smaller and larger fragments obtained by splitting the fragment of mass ξ1\xi_1, similarly for ξ2\xi_2). Find the density of:

?
(a)

ξ1\xi_1;

(b)

ξ11\xi_{11}.

Problem 11.33

Let the independent random variables ξ1,…,ξn\xi_1, \ldots , \xi_n be uniformly distributed on the segment [0,1][0, 1]. Find the distribution of the product η=∏i=1nξi\eta = \prod_{i=1}^n \xi_i.

?
Problem 11.34

The random variables ξ\xi and η\eta are independent and have the same distribution with density

pξ(x)=pη(x)=c1+x4. p_\xi (x) = p_\eta (x) = \frac{c}{1 + x^4}.

Find the constant cc and prove that the quantity ξ/η\xi /\eta has the Cauchy distribution.

?
Problem 11.35

Let ξ\xi and η\eta be independent random variables, with P{ξ=0}=P{ξ=1}=1/2\mathbb {P}\left\{ \xi = 0\right\} = \mathbb {P}\left\{ \xi = 1\right\} = 1/2, and P{η<x}=x\mathbb {P}\left\{ \eta < x\right\} = x for x∈[0,1]x \in [0, 1]. Find the distribution functions of the following random variables:

?
(a)

ξ+η\xi + \eta;

(b)

ξ/2+η\xi /2 + \eta;

(c)

ξ+η/2\xi + \eta /2;

(d)

ξη\xi \eta;

(e)

ηξ\eta^\xi;

(f)

∣ξ−η∣\left|\xi - \eta \right|.

Plot the graphs of the distribution functions found.

Problem 11.36

Let the random variables ξ\xi and η\eta be independent, with ξ\xi having an exponential distribution with parameter α\alpha and η\eta uniform on the segment [0,h][0, h]. Find the densities of:

?
(a)

the sum ξ+η\xi + \eta;

(b)

the difference ξ−η\xi - \eta.

Problem 11.37

The independent random variables ξ\xi, η\eta, and ζ\zeta have the same exponential distribution with parameter α\alpha. Find the densities of the random variables:

?
(a)

ξ+ηξ\dfrac {\xi + \eta }{\xi };

(b)

ξ+ηζ\dfrac {\xi + \eta }{\zeta }.

Problem 11.38

Let ξ1,ξ2,…\xi_1, \xi_2, \ldots be independent identically distributed random variables, with ξ1\xi_1 taking the values 0,1,…,90, 1, \ldots , 9 with probability 1/101/10 each. Find the distribution of the sum of the series

∑i=1∞ξi10i. \sum _{i=1}^\infty \frac{\xi _i}{10^i}.
?
Problem 11.39

Let ξ1,ξ2,…\xi_1, \xi_2, \ldots be independent identically distributed random variables, with ξ1\xi_1 taking the values 0 and 1 with probability 1/21/2 each. Find the distributions of the sums of the following series:

?
(a)

∑i=1∞ξi2i\displaystyle \sum_{i=1}^\infty \frac{\xi_i}{2^i};

(b)

23∑i=1∞ξi3i−1\dfrac {2}{3}\displaystyle \sum_{i=1}^\infty \frac{\xi_i}{3^{i-1}}.

Problem 11.40

Let the independent random variables ξ\xi and η\eta have the same exponential distribution with parameter α\alpha. Set U=min⁡(ξ,η)U = \min (\xi , \eta ) and:

?
(a)

V=ξ−ηV = \xi - \eta;

(b)

V=  1ξ<ηV = \; \mathbb {1}_{\xi < \eta }.

Prove that the random variables UU and VV are independent.

Problem 11.41

Generalize the result of part (b) of problem 11.40 to an arbitrary continuous distribution.

?
Problem 11.42

The random variables ξ\xi and η\eta are independent and have an exponential distribution with parameter 1. Prove that the ratio ξξ+η\dfrac {\xi }{\xi + \eta } is uniformly distributed on the segment [0,1][0, 1].

?
Problem 11.43

Let the independent random variables ξ\xi and η\eta have a normal distribution with parameters α\alpha and σ2\sigma^2. Find the probability density of the random variable

ξ+η−2αξ−η. \frac{\xi + \eta - 2\alpha }{\xi - \eta }.
?
Problem 11.44

Let ξ1,…,ξn\xi_1, \ldots , \xi_n be independent identically distributed random variables with continuous distribution function FF. Denote by νn(x)\nu_n(x) the number of the ξi\xi_i that are less than xx. Prove that the distribution of the random variable

Dn=sup⁡x∣νn(x)n−F(x)∣ D_n = \sup _x \left|\frac{\nu _n(x)}{n} - F(x)\right|

does not depend on FF.

?
Problem 11.45

Prove that the integral equation

∫−∞∞x(t) dt(y−t)2+1=e−y2 \int _{-\infty }^{\infty } \frac{x(t)\, dt}{(y-t)^2 + 1} = e^{-y^2}

has no solution in the class of nonnegative functions.

?
Section 12Moments[62/66%]
Problem 12.1

Suppose a random variable has:

?
(a)

a binomial distribution;

(b)

a Poisson distribution with parameter λ\lambda;

(c)

a geometric distribution with parameter pp;

(d)

a uniform distribution on the segment [−a,a][-a, a];

(e)

a uniform distribution on the segment [a,b][a, b];

(f)

a Cauchy distribution with location parameter aa;

(g)

an exponential distribution with parameter α\alpha;

(h)

a Laplace distribution;

(i)

a normal distribution with parameters aa and σ2\sigma^2.

Does it have an expectation and a variance? If so, compute the corresponding values.

Problem 12.2

A discrete random variable ξ\xi has the distribution series

xi−1012P{ξ=xi}1/51/103/102/5 \begin{array}{c|cccc} x_i & -1 & 0 & 1 & 2 \\ \hline \mathbb {P}\left\{ \xi = x_i\right\} & 1/5 & 1/10 & 3/10 & 2/5 \end{array}

Find the expectations and variances of the random variables:

?
(a)

ξ\xi;

(b)

∣ξ∣\left|\xi \right|;

(c)

ξ2\xi^2;

(d)

2ξ2\xi.

Problem 12.3

Solve problem 12.2 if the discrete random variable ξ\xi has the distribution series

xi−101P{ξ=xi}1/31/61/2 \begin{array}{c|ccc} x_i & -1 & 0 & 1 \\ \hline \mathbb {P}\left\{ \xi = x_i\right\} & 1/3 & 1/6 & 1/2 \end{array}
?
(a)
(b)
(c)
(d)
Problem 12.4

Let the random variable ξ\xi take the values −2-2, −1-1, 00, 11, and 22 with equal probabilities. Find the expectations and variances of the random variables:

?
(a)

ξ\xi;

(b)

−ξ-\xi;

(c)

∣ξ∣\left|\xi \right|;

(d)

ξ2\xi^2.

Problem 12.5

The two-dimensional distribution of a pair of integer-valued random variables ξ\xi and η\eta is given by the table

η\ξ−101−11/81/127/2415/241/61/8 \begin{array}{c|ccc} {}_\eta \backslash ^\xi & -1 & 0 & 1 \\ \hline -1 & 1/8 & 1/12 & 7/24 \\ 1 & 5/24 & 1/6 & 1/8 \end{array}

where the entry at the intersection of column ξ=i\xi = i and row η=j\eta = j is the probability P{ξ=i,η=j}\mathbb {P}\left\{ \xi = i, \eta = j\right\}. Find:

?
(a)

E[ξ]\mathbb {E}\left[\xi \right], Var⁡[ξ]\operatorname {Var}\left[\xi \right];

(b)

E[η]\mathbb {E}\left[\eta \right], Var⁡[η]\operatorname {Var}\left[\eta \right];

(c)

E[ξη]\mathbb {E}\left[\xi \eta \right];

(d)

E[ξ−2η]\mathbb {E}\left[\xi - 2\eta \right], Var⁡[ξ−2η]\operatorname {Var}\left[\xi - 2\eta \right].

Problem 12.6

Solve problem 12.5 if the two-dimensional distribution of the random variables ξ\xi and η\eta is given by the table

η\ξ−201−11/61/61/621/61/61/6 \begin{array}{c|ccc} {}_\eta \backslash ^\xi & -2 & 0 & 1 \\ \hline -1 & 1/6 & 1/6 & 1/6 \\ 2 & 1/6 & 1/6 & 1/6 \end{array}
?
(a)
(b)
(c)
Problem 12.7

The probability of hitting a target on each shot is constant and equal to 3/53/5. Three shots are fired. Compute the expectation and variance of the number of hits.

?
Problem 12.8

The probability of hitting a target on each shot is constant and equal to 3/53/5. Shots are fired until the first hit. Compute the expectation and variance of the number of shots fired, if the shooter has only 4 rounds.

?
Problem 12.9

A discrete random variable ξ\xi has the distribution series

xi01x3P{ξ=xi}p10,30,6 \begin{array}{c|ccc} x_i & 0 & 1 & x_3 \\ \hline \mathbb {P}\left\{ \xi = x_i\right\} & p_1 & 0{,}3 & 0{,}6 \end{array}

Find the probability p1p_1 and the value x3x_3, given that E[ξ]=2,7\mathbb {E}\left[\xi \right] = 2{,}7.

?
Problem 12.10

The random variable ξ\xi can take only the following values: −2-2, −1-1, 00, 11, and 22. With what probabilities does ξ\xi take these values, if:

?
(a)

E[ξ]=E[ξ3]=0\mathbb {E}\left[\xi \right] = \mathbb {E}\left[\xi^3\right] = 0, E[ξ2]=1\mathbb {E}\left[\xi^2\right] = 1, and E[ξ4]=2\mathbb {E}\left[\xi^4\right] = 2;

(b)

E[ξ]=E[ξ3]=0\mathbb {E}\left[\xi \right] = \mathbb {E}\left[\xi^3\right] = 0, E[ξ2]=2\mathbb {E}\left[\xi^2\right] = 2, and E[ξ4]=6\mathbb {E}\left[\xi^4\right] = 6?

Problem 12.11

The random variable ξ\xi has a binomial distribution. Determine the parameters of the distribution, given that E[ξ]=3,2\mathbb {E}\left[\xi \right] = 3{,}2 and σ2=0,64\sigma^2 = 0{,}64.

?
Problem 12.12

Compute E[(1+ξ)−1]\mathbb {E}\left[(1+\xi )^{-1}\right], if the random variable ξ\xi has:

?
(a)

a Poisson distribution with parameter λ\lambda;

(b)

a binomial distribution with parameters nn and pp.

Problem 12.13

An electron in a force field has energy VV taking the values vi=a(i+1/2)v_i = a(i + 1/2), i=1,2,…i = 1, 2, \ldots, with probabilities proportional to e−bvie^{-bv_i}; aa and bb are constants. Find E[V]\mathbb {E}\left[V\right] and Var⁡[V]\operatorname {Var}\left[V\right].

?
Problem 12.14

Let ξ1,ξ2,…\xi_1, \xi_2, \ldots be independent random variables taking the values 1 and 0 with probabilities pp and 1−p1-p respectively. Find the mean value and variance of the sum ζ=η1+…+ηn\zeta = \eta_1 + \ldots + \eta_n, if:

?
(a)

ηi=ξiξi+1\eta_i = \xi_i \xi_{i+1};

(b)

ηi={0,if ξi+ξi+1 is even,1,if ξi+ξi+1=1.\eta_i = \begin{cases} 0, & \text{if } \xi_i + \xi_{i+1} \text{ is even}, \\ 1, & \text{if } \xi_i + \xi_{i+1} = 1. \end{cases}

Problem 12.15

nn dice are rolled. Find the expectation and variance of the sum of the pips on all the dice.

?
Problem 12.16

For a group of nn people, find the expectation of the number of days in the year on each of which:

?
(a)

the birthdays of exactly kk people fall;

(b)

the birthdays of at least two people fall.

Problem 12.17

Prove that if the random variable ξ\xi takes only nonnegative integer values, then

E[ξ]=∑n=1∞P{ξ⩾n}. \mathbb {E}\left[\xi \right] = \sum _{n=1}^\infty \mathbb {P}\left\{ \xi \geqslant n\right\} .
?
Problem 12.18

Let rr numbered particles be placed at random into nn cells. Find the mean value of the number of empty cells.

?
Problem 12.19

The independent random variables ξ1,…,ξn\xi_1, \ldots , \xi_n are uniformly distributed on the set {1,2,…,J}\left\{ 1, 2, \ldots , J\right\}. Define the random variable ν\nu to be the number of distinct values among ξ1,ξ2,…,ξn\xi_1, \xi_2, \ldots , \xi_n. Find E[ν]\mathbb {E}\left[\nu \right].

?
Problem 12.20

Prove that if the random variable ξ\xi takes only nonnegative values, then

∑n=1∞P{ξ⩾n}⩽E[ξ]⩽∑n=1∞P{ξ⩾n}+1. \sum _{n=1}^\infty \mathbb {P}\left\{ \xi \geqslant n\right\} \leqslant \mathbb {E}\left[\xi \right] \leqslant \sum _{n=1}^\infty \mathbb {P}\left\{ \xi \geqslant n\right\} + 1.
?
Problem 12.21
?
(a)

Prove that E[∣ξ∣]\mathbb {E}\left[\left|\xi \right|\right] is finite if and only if ∑nP{∣ξ∣>n}<∞\displaystyle \sum_n \mathbb {P}\left\{ \left|\xi \right| > n\right\} < \infty.

(b)

Prove that E[ξ2]\mathbb {E}\left[\xi^2\right] is finite if and only if ∑nnP{∣ξ∣>n}<∞\displaystyle \sum_n n\mathbb {P}\left\{ \left|\xi \right| > n\right\} < \infty.

Problem 12.22
?
(a)

Prove that if P{ξ⩾0}=1\mathbb {P}\left\{ \xi \geqslant 0\right\} = 1, then

E[ξ]=∫0∞P{ξ⩾x} dx. \mathbb {E}\left[\xi \right] = \int _0^\infty \mathbb {P}\left\{ \xi \geqslant x\right\} \, dx.
(b)

Prove that if E[∣ξ∣]<∞\mathbb {E}\left[\left|\xi \right|\right] < \infty, then

E[ξ]=∫0∞P{ξ⩾x} dx−∫−∞0P{ξ<x} dx. \mathbb {E}\left[\xi \right] = \int _0^\infty \mathbb {P}\left\{ \xi \geqslant x\right\} \, dx - \int _{-\infty }^0 \mathbb {P}\left\{ \xi < x\right\} \, dx.
Problem 12.23

Let the independent random variables ξ1,ξ2,…\xi_1, \xi_2, \ldots have the same distribution function FF. Let xx be a number such that 0<F(x)<10 < F(x) < 1. Denote by ν(x)\nu (x) the index of the first random variable whose value is greater than or equal to xx. Prove that E[ν(x)]=(1−F(x))−1\mathbb {E}\left[\nu (x)\right] = (1 - F(x))^{-1}.

?
Problem 12.24

Let ξ1,ξ2,…\xi_1, \xi_2, \ldots be independent identically distributed random variables with mean aa and variance σ2\sigma^2. Let ν\nu be a positive integer-valued random variable, independent of {ξn}\left\{ \xi_n\right\}, with mean bb and variance θ2\theta^2. Find the mean value and variance of the random sum ξ1+…+ξν\xi_1 + \ldots + \xi_\nu.

?
Problem 12.25

The distribution of the random variable ξ\xi has density

p(x)={2xfor x∈[0,1],0for x∉[0,1]. p(x) = \begin{cases} 2x & \text{for } x \in [0, 1], \\ 0 & \text{for } x \notin [0, 1]. \end{cases}

Find the expectations and variances of the following quantities:

?
(a)

ξ\xi;

(b)

∣ξ−1/2∣\left|\xi - 1/2\right|;

(c)

ξ2\xi^2;

(d)

ξ2+ξ\xi^2 + \xi.

Problem 12.26

The distribution of the random variable ξ\xi has density

p(x)={1/x2for x⩾1,0for x<1. p(x) = \begin{cases} 1/x^2 & \text{for } x \geqslant 1, \\ 0 & \text{for } x < 1. \end{cases}

Find the expectations and variances of the following quantities:

?
(a)

ξ\xi;

(b)

1/ξ1/\xi.

Problem 12.27

The diameter of a circle is measured approximately. Assuming its value is uniformly distributed on the segment [a,b][a, b], find the mean value and variance of the area of the circle.

?
Problem 12.28

Let the random variable ξ\xi be uniformly distributed on the segment [0,1][0, 1]. Compute E[sin⁡2πξ]\mathbb {E}\left[\sin^2\pi \xi \right], Var⁡[sin⁡2πξ]\operatorname {Var}\left[\sin^2\pi \xi \right].

?
Problem 12.29

Let the random variable ξ\xi have the standard Cauchy distribution. Compute E[min⁡(∣ξ∣,1)]\mathbb {E}\left[\min (\left|\xi \right|, 1)\right].

?
Problem 12.30

The random variable ξ\xi has an exponential distribution with parameter α\alpha. Determine for which values of α\alpha the expectation and variance of the random variable eξe^\xi exist.

?
Problem 12.31

The random variable ξ\xi has the standard normal distribution. Find:

?
(a)

E[∣ξ∣]\mathbb {E}\left[\left|\xi \right|\right];

(b)

E[ξn]\mathbb {E}\left[\xi^n\right], n∈Nn \in \mathbb {N}.

Problem 12.32

The random variable ξ\xi has a normal distribution with parameters aa and σ2\sigma^2. Find E[∣ξ−E[ξ]∣]\mathbb {E}\left[\left|\xi - \mathbb {E}\left[\xi \right]\right|\right].

?
Problem 12.33

In addition to the conditions of problem 6.22, suppose the match organizers have set up a prize fund of 100 tugriks. What sum of money should be awarded to player A?

?
Problem 12.34

The random variables ξ\xi and η\eta are independent, with ξ\xi having a normal distribution with parameters 2 and 1/21/2, and η\eta a uniform distribution on the segment [0,4][0, 4]. Find:

?
(a)

E[ξ+η]\mathbb {E}\left[\xi + \eta \right];

(b)

E[ξη]\mathbb {E}\left[\xi \eta \right];

(c)

E[ξ2]\mathbb {E}\left[\xi^2\right];

(d)

E[ξ−η2]\mathbb {E}\left[\xi - \eta^2\right];

(e)

Var⁡[ξ+η]\operatorname {Var}\left[\xi + \eta \right];

(f)

Var⁡[ξ−η]\operatorname {Var}\left[\xi - \eta \right].

Problem 12.35

The random variables ξ\xi and η\eta are independent, with E[ξ]=2\mathbb {E}\left[\xi \right] = 2, Var⁡[ξ]=1\operatorname {Var}\left[\xi \right] = 1, E[η]=1\mathbb {E}\left[\eta \right] = 1, Var⁡[η]=3\operatorname {Var}\left[\eta \right] = 3. Find the mean value and variance of the random variable:

?
(a)

ξ−2η\xi - 2\eta;

(b)

2ξ−η2\xi - \eta.

Problem 12.36

The random variables ξ\xi, η\eta, and ζ\zeta are independent and have the standard normal distribution. Find the mean value and variance of the random variable ξ+ξη+ξηζ\xi + \xi \eta + \xi \eta \zeta.

?
Problem 12.37

Prove that if E[ξ2]=E[ξ3]=E[ξ4]\mathbb {E}\left[\xi^2\right] = \mathbb {E}\left[\xi^3\right] = \mathbb {E}\left[\xi^4\right], then the random variable ξ\xi is discrete and can take only the values 0 and 1.

?
Problem 12.38

Determine whether there exists a random variable ξ\xi such that E[ξ]=1\mathbb {E}\left[\xi \right] = 1, E[ξ2]=2\mathbb {E}\left[\xi^2\right] = 2, E[ξ3]=3\mathbb {E}\left[\xi^3\right] = 3, E[ξ4]=4\mathbb {E}\left[\xi^4\right] = 4, and E[ξ5]=5\mathbb {E}\left[\xi^5\right] = 5?

?
Problem 12.39

Find a possible distribution of the random variable ξ\xi, if E[ξ2n]\mathbb {E}\left[\xi^{2n}\right], E[ξ2n+1]\mathbb {E}\left[\xi^{2n+1}\right], and E[ξ2n+2]\mathbb {E}\left[\xi^{2n+2}\right] for some natural nn are consecutive terms of an arithmetic progression.

?
Problem 12.40

Find a possible distribution of the random variable ξ\xi, if E[ξn]\mathbb {E}\left[\xi^n\right], E[ξn+1]\mathbb {E}\left[\xi^{n+1}\right], and E[ξn+2]\mathbb {E}\left[\xi^{n+2}\right] for some natural nn are consecutive terms of a geometric progression.

?
Problem 12.41

A random point (ξ,η)(\xi , \eta ) has a uniform distribution in the square {(x,y):0⩽x,y⩽1}\left\{ (x, y) : 0 \leqslant x, y \leqslant 1\right\}. Find the mean values and variances of the following quantities:

?
(a)

ζ=ξη\zeta = \xi \eta;

(b)

ζ=max⁡(ξ,η)\zeta = \max (\xi , \eta ).

Problem 12.42

Let the random variables ξ\xi and η\eta be independent, with ξ\xi uniformly distributed on the segment [0,1][0, 1] and η\eta on the segment [1,2][1, 2]. Find the mean values and variances of the following quantities:

?
(a)

ζ=ξη\zeta = \xi \eta;

(b)

ζ=max⁡(ξ,η)\zeta = \max (\xi , \eta ).

Problem 12.43

Give an example of a random variable ξ\xi:

?
(a)

with a discrete;

(b)

with an absolutely continuous

distribution for which the mean value does not exist.

Problem 12.44

Give an example of a random variable ξ\xi with:

?
(a)

a discrete;

(b)

an absolutely continuous

distribution for which the mean value exists but the variance does not.

Problem 12.45

Give an example of a random variable ξ\xi:

?
(a)

with a discrete;

(b)

with an absolutely continuous

distribution for which the kk-th moment exists but the (k+1)(k+1)-th does not.

Problem 12.46

Give an example of a nonnegative random variable ξ\xi for which

?
(a)

E[ξ]\mathbb {E}\left[\xi \right] is the median;

(b)

E[ξ]\mathbb {E}\left[\xi \right] is strictly greater than the median;

(c)

E[ξ]\mathbb {E}\left[\xi \right] is strictly less than the median.

Problem 12.47

Find:

?
(a)

inf⁡a∈RE[(ξ−a)2]\inf_{a \in \mathbb {R}} \mathbb {E}\left[(\xi - a)^2\right];

(b)

inf⁡a∈RE[∣ξ−a∣]\inf_{a \in \mathbb {R}} \mathbb {E}\left[\left|\xi - a\right|\right].

Problem 12.48

Give an example of dependent random variables ξ\xi and η\eta for which E[ξη]=E[ξ]E[η]\mathbb {E}\left[\xi \eta \right] = \mathbb {E}\left[\xi \right]\mathbb {E}\left[\eta \right].

?
Problem 12.49

In a certain Western country a boy earns extra money selling newspapers; he buys a newspaper for 4 cents each and sells it for 7 cents. He cannot return unsold newspapers and loses money on them. Suppose the number of possible sales in a day is random and has a Poisson distribution with mean 50. How many newspapers should the poor fellow buy in order to maximize his mean profit?

?
Problem 12.50

Let the random variables ξ\xi and η\eta have densities ff and gg respectively; let aa be some number. Prove that if f(x)⩾g(x)f(x) \geqslant g(x) for x<ax < a and f(x)⩽g(x)f(x) \leqslant g(x) for x>ax > a, then E[ξ]⩽E[η]\mathbb {E}\left[\xi \right] \leqslant \mathbb {E}\left[\eta \right].

?
Problem 12.51

Let the random variables ξ1\xi_1 and ξ2\xi_2 be positive and identically distributed. Is it true that

E[ξ1ξ1+ξ2]=E[ξ2ξ1+ξ2] ? \mathbb {E}\left[\frac{\xi _1}{\xi _1+\xi _2}\right] = \mathbb {E}\left[\frac{\xi _2}{\xi _1+\xi _2}\right]\, ?
?
Problem 12.52

Prove that if the random variables ξ1,…,ξn\xi_1, \ldots , \xi_n are independent, positive, and identically distributed, then for k⩽nk \leqslant n

E[ξ1+…+ξkξ1+…+ξn]=kn. \mathbb {E}\left[\frac{\xi _1 + \ldots + \xi _k}{\xi _1 + \ldots + \xi _n}\right] = \frac{k}{n}.
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Problem 12.53

Prove that if each of the random variables ξ\xi and η\eta takes only two values, then the equality E[ξη]=E[ξ]E[η]\mathbb {E}\left[\xi \eta \right] = \mathbb {E}\left[\xi \right]\mathbb {E}\left[\eta \right] implies the independence of ξ\xi and η\eta.

?
Problem 12.54

Let the random variable ξ\xi be such that P{0<ξ<1}=1\mathbb {P}\left\{ 0 < \xi < 1\right\} = 1. Prove that Var⁡[ξ]<E[ξ]\operatorname {Var}\left[\xi \right] < \mathbb {E}\left[\xi \right].

?
Problem 12.55

What condition must the independent random variables ξ\xi and η\eta satisfy for Var⁡[ξη]=Var⁡[ξ]Var⁡[η]\operatorname {Var}\left[\xi \eta \right] = \operatorname {Var}\left[\xi \right]\operatorname {Var}\left[\eta \right]?

?
Problem 12.56

Let ξ\xi be a random variable such that E[∣ξ−1∣3]\mathbb {E}\left[\left|\xi -1\right|^3\right] exists. Prove that E[∣ξ∣3]\mathbb {E}\left[\left|\xi \right|^3\right] exists.

?
Problem 12.57

Let ξ\xi and η\eta be independent random variables such that E[∣ξ+η∣a]\mathbb {E}\left[\left|\xi +\eta \right|^a\right] exists, a>0a > 0. Prove that E[∣ξ∣a]\mathbb {E}\left[\left|\xi \right|^a\right] and E[∣η∣a]\mathbb {E}\left[\left|\eta \right|^a\right] exist.

?
Problem 12.58

Let ξ\xi be a random variable and a>1a > 1 a real number. Prove that if

lim⁡x→∞P{∣ξ∣>ax}P{∣ξ∣>x}=0, \lim _{x \to \infty } \frac{\mathbb {P}\left\{ \left|\xi \right| > ax\right\} }{\mathbb {P}\left\{ \left|\xi \right| > x\right\} } = 0,

then E[∣ξ∣k]\mathbb {E}\left[\left|\xi \right|^k\right] exists for every k>0k > 0.

?
Problem 12.59

Let the random variable ξ\xi take a finite number of nonnegative values x1,…,xsx_1, \ldots , x_s. Prove that as n→∞n \to \infty:

?
(a)

E[ξn+1]E[ξn]→max⁡(x1,…,xs)\dfrac {\mathbb {E}\left[\xi^{n+1}\right]}{\mathbb {E}\left[\xi^n\right]} \to \max (x_1, \ldots , x_s);

(b)

E[ξn]n→max⁡(x1,…,xs)\sqrt[n]{\mathbb {E}\left[\xi^n\right]} \to \max (x_1, \ldots , x_s).

Problem 12.60

Let ξ\xi and η\eta be random variables. Prove that if E[ξ]\mathbb {E}\left[\xi \right] and E[η]\mathbb {E}\left[\eta \right] exist, then E[max⁡(ξ,η)]\mathbb {E}\left[\max (\xi , \eta )\right] exists. Is the converse true? Is the converse true for nonnegative random variables ξ\xi and η\eta?

?
Problem 12.61

Let ξ\xi and η\eta be random variables. Prove that if E[max⁡(ξ,η)]\mathbb {E}\left[\max (\xi , \eta )\right] and E[min⁡(ξ,η)]\mathbb {E}\left[\min (\xi , \eta )\right] exist, then E[ξ]\mathbb {E}\left[\xi \right] and E[η]\mathbb {E}\left[\eta \right] exist, and E[ξ]+E[η]=E[max⁡(ξ,η)]+E[min⁡(ξ,η)]\mathbb {E}\left[\xi \right] + \mathbb {E}\left[\eta \right] = \mathbb {E}\left[\max (\xi , \eta )\right] + \mathbb {E}\left[\min (\xi , \eta )\right].

?
Problem 12.62

Let the positive random variables ξ\xi and η\eta be independent with densities ff and gg respectively. Prove that if E[ξ]<∞\mathbb {E}\left[\xi \right] < \infty, then the ratio ξ/η\xi /\eta has a finite expectation if and only if

∫01g(y)y dy<∞. \int _0^1 \frac{g(y)}{y}\, dy < \infty .
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Section 13Covariance and the Correlation Coefficient[25/68%]
Problem 13.1

Prove that

Var⁡[ξ1+…+ξn]=∑i=1nVar⁡[ξi]+∑i≠jCov⁡[ξi,ξj]. \operatorname {Var}\left[\xi _1 + \ldots + \xi _n\right] = \sum _{i=1}^n \operatorname {Var}\left[\xi _i\right] + \sum _{i \neq j} \operatorname {Cov}\left[ \xi _i, \xi _j \right].
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Problem 13.2

Prove that independence of the random variables ξ\xi and η\eta implies Cov⁡[ξ,η]=0\operatorname {Cov}\left[ \xi , \eta \right] = 0.

?
Problem 13.3

Prove that:

?
(a)

∣Corr⁡[ξ,η]∣⩽1\left|\operatorname {Corr}\left[ \xi , \eta \right]\right| \leqslant 1;

(b)

∣Corr⁡[ξ,η]∣=1\left|\operatorname {Corr}\left[ \xi , \eta \right]\right| = 1 if and only if the random variables ξ\xi and η\eta are linearly dependent with probability 1, i.e., there exist constants a≠0a \neq 0 and bb such that P{η=aξ+b}=1\mathbb {P}\left\{ \eta = a\xi + b\right\} = 1.

Problem 13.4

Construct an example of two dependent random variables whose correlation coefficient is zero.

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Problem 13.5

Find Cov⁡[ξ,η]\operatorname {Cov}\left[ \xi , \eta \right] for the random variables with the distribution table given in problem:

?
(a)

12.5;

(b)

12.6.

Problem 13.6

Let the random variable ξ\xi take the values 0, 1, and −1-1 with probabilities 1/31/3, 1/21/2, and 1/61/6 respectively. Find the correlation coefficient between ξ\xi and ξ2\xi^2.

?
Problem 13.7

Let the random variables ξ\xi and η\eta be independent, with P{ξ=1}=P{ξ=−1}=1/2\mathbb {P}\left\{ \xi =1\right\} = \mathbb {P}\left\{ \xi =-1\right\} = 1/2, P{η=1}=P{η=−1}=1/4\mathbb {P}\left\{ \eta =1\right\} = \mathbb {P}\left\{ \eta =-1\right\} = 1/4, and P{η=0}=1/2\mathbb {P}\left\{ \eta =0\right\} = 1/2. Are the random variables ξη\xi \eta and η\eta:

?
(a)

independent;

(b)

uncorrelated?

Problem 13.8

Find the correlation coefficient between the number of «ones» rolled and the number of «sixes» rolled in nn rolls of a fair die.

?
Problem 13.9

A random point (ξ,η)(\xi , \eta ) has a uniform distribution in the square {(x,y):∣x∣+∣y∣⩽2}\left\{ (x, y) : \left|x\right| + \left|y\right| \leqslant \sqrt{2}\right\}.

?
(a)

Find the mean value and variance of ξ\xi.

(b)

Find the mean value and variance of η\eta.

(c)

Find Cov⁡[ξ,η]\operatorname {Cov}\left[ \xi , \eta \right].

(d)

Determine whether ξ\xi and η\eta are dependent.

Problem 13.10

A random point (ξ,η)(\xi , \eta ) has a uniform distribution in the triangle with vertices at the points (0,0)(0, 0), (2,1)(2, 1), (2,0)(2, 0).

?
(a)

Find the mean value and variance of ξ\xi.

(b)

Find the mean value and variance of η\eta.

(c)

Find Cov⁡[ξ,η]\operatorname {Cov}\left[ \xi , \eta \right].

(d)

Determine whether ξ\xi and η\eta are dependent.

Problem 13.11

A random point (ξ,η)(\xi , \eta ) is uniformly distributed inside the unit disk centered at the origin.

?
(a)

Find the expectation and variance of ζ=ξη\zeta = \xi \eta.

(b)

Find Cov⁡[ξ,η]\operatorname {Cov}\left[ \xi , \eta \right].

(c)

Determine whether ξ\xi and η\eta are dependent.

Problem 13.12

The joint distribution of the random variables ξ\xi and η\eta has density

p(x,y)={x+yfor 0⩽x⩽1, 0⩽y⩽1,0otherwise. p(x, y) = \begin{cases} x+y & \text{for } 0 \leqslant x \leqslant 1,\ 0 \leqslant y \leqslant 1, \\ 0 & \text{otherwise.} \end{cases}
?
(a)

Find the mean value and variance of ξ\xi.

(b)

Find the mean value and variance of η\eta.

(c)

Find Cov⁡[ξ,η]\operatorname {Cov}\left[ \xi , \eta \right].

Problem 13.13

Two points are thrown at random onto the segment [0,a][0, a]. Find:

?
(a)

the expectation and variance of the distance between them;

(b)

the covariance of the coordinates of the left and right points.

Problem 13.14

Find the correlation coefficient between the random variables ξ\xi and η=e−ξ\eta = e^{-\xi }, if ξ\xi has the standard normal distribution.

?
Problem 13.15

The random variables ξ\xi and η\eta are independent and normally distributed with parameters aa and σ2\sigma^2. Find:

?
(a)

the correlation coefficient of the quantities αξ+βη\alpha \xi + \beta \eta and αξ−βη\alpha \xi - \beta \eta;

(b)

their joint distribution.

Problem 13.16

Let ξ\xi be a random variable with a symmetric distribution and finite variance. Find the correlation coefficient of the random variables ξ\xi and ∣ξ∣\left|\xi \right|.

?
Problem 13.17

Let the random variables ξ1,…,ξn+m\xi_1, \ldots , \xi_{n+m} be independent, identically distributed, and have finite variance. For n>mn > m, find the correlation coefficient between the sums ξ1+…+ξn\xi_1 + \ldots + \xi_n and ξm+1+…+ξm+n\xi_{m+1} + \ldots + \xi_{m+n}.

?
Problem 13.18

Give an example of two random variables ξ\xi and η\eta such that E[ξ]\mathbb {E}\left[\xi \right] and E[η]\mathbb {E}\left[\eta \right] exist, but E[ξη]\mathbb {E}\left[\xi \eta \right] does not exist.

?
Problem 13.19

An urn contains aa white and bb black balls. kk balls are drawn from the urn simultaneously, k⩽a+bk \leqslant a+b. Find the expectation and variance of the number of white balls drawn.

?
Problem 13.20

For a group of nn people, find the expectation and variance of the number of days in the year on which no one's birthday falls.

?
Problem 13.21

nn letters are written, addressed to different recipients. There are nn envelopes with the corresponding addresses. The letters are placed into the envelopes in a random order. Find the mean value and variance of the number of letters sent to the correct address.

?
Problem 13.22

Prove that there do not exist three random variables ξ\xi, η\eta, and ζ\zeta such that the correlation coefficient of any two of them equals −1-1.

?
Problem 13.23

Let ξ1,…,ξn\xi_1, \ldots , \xi_n be random variables such that the correlation coefficient of any two of them equals the same number ρ\rho. Prove that ρ⩾−1/(n−1)\rho \geqslant -1/(n-1).

?
Problem 13.24

Let the random variables ξ\xi and η\eta have zero mean values, unit variances, and correlation coefficient ρ\rho. Show that E[max⁡(ξ2,η2)]⩽1+1−ρ2\mathbb {E}\left[\max (\xi^2, \eta^2)\right] \leqslant 1 + \sqrt{1-\rho^2}.

?
Problem 13.25

Let the random variables ξ\xi and η\eta have zero mean values, unit variances, and correlation coefficient ρ\rho, 0<ρ<10 < \rho < 1. Find all values of aa and bb for which the random variables ξ−aη\xi - a\eta and η−bξ\eta - b\xi are uncorrelated.

?
Section 14Inequalities[16/25%]
Problem 14.1

Let g(x)g(x) be a nonnegative increasing function and ξ\xi an arbitrary random variable. Prove that for any xx the inequality

P{ξ⩾x}⩽E[g(ξ)]g(x) \mathbb {P}\left\{ \xi \geqslant x\right\} \leqslant \frac{\mathbb {E}\left[g(\xi )\right]}{g(x)}

holds.

?
Problem 14.2

Let E[ξ2]<∞\mathbb {E}\left[\xi^2\right] < \infty. Prove that

P{∣ξ−E[ξ]∣>3Var⁡[ξ]}⩽1/9. \mathbb {P}\left\{ \left|\xi - \mathbb {E}\left[\xi \right]\right| > 3\sqrt{\operatorname {Var}\left[\xi \right]}\right\} \leqslant 1/9.
?
Problem 14.3

Prove that for arbitrary random variables ξ\xi and η\eta

E[∣ξη∣]⩽E[ξ2]E[η2]. \mathbb {E}\left[\left|\xi \eta \right|\right] \leqslant \sqrt{\mathbb {E}\left[\xi ^2\right]\mathbb {E}\left[\eta ^2\right]}.
?
Problem 14.4

Let E[∣ξ∣r]<∞\mathbb {E}\left[\left|\xi \right|^r\right] < \infty. Prove that:

?
(a)

(E[∣ξ∣s])r⩽(E[∣ξ∣r])s(\mathbb {E}\left[\left|\xi \right|^s\right])^r \leqslant (\mathbb {E}\left[\left|\xi \right|^r\right])^s for any s∈(0,r)s \in (0, r);

(b)

(E[∣ξ∣s])1/s⩽(E[∣ξ∣r])1/r(\mathbb {E}\left[\left|\xi \right|^s\right])^{1/s} \leqslant (\mathbb {E}\left[\left|\xi \right|^r\right])^{1/r} for any s∈(0,r)s \in (0, r).

Problem 14.5

Prove that E[∣ξ∣]⩽E[∣ξ∣r]r\mathbb {E}\left[\left|\xi \right|\right] \leqslant \sqrt[r]{\mathbb {E}\left[\left|\xi \right|^r\right]} for r>1r > 1.

?
Problem 14.6

Prove that for a random variable ξ\xi with finite variance and any x>0x > 0

P{∣ξ−E[ξ]∣>x}⩽2Var⁡[ξ]x2+Var⁡[ξ]. \mathbb {P}\left\{ \left|\xi - \mathbb {E}\left[\xi \right]\right| > x\right\} \leqslant \frac{2\operatorname {Var}\left[\xi \right]}{x^2 + \operatorname {Var}\left[\xi \right]}.
?
Problem 14.7

Let ξ\xi be a random variable with E[ehξ]<∞\mathbb {E}\left[e^{h\xi }\right] < \infty, h>0h > 0. Prove that for any xx

P{ξ⩾x}⩽E[ehξ]/ehx. \mathbb {P}\left\{ \xi \geqslant x\right\} \leqslant \mathbb {E}\left[e^{h\xi }\right]/e^{hx}.
?
Problem 14.8

Let the random variable ξ\xi have a Poisson distribution with parameter λ\lambda. Prove that for any x>0x > 0:

?
(a)

P{ξ⩾x}⩽λ/x\mathbb {P}\left\{ \xi \geqslant x\right\} \leqslant \lambda /x;

(b)

P{ξ⩾x}⩽eλ/2x\mathbb {P}\left\{ \xi \geqslant x\right\} \leqslant e^\lambda /2^x;

(c)

P{ξ⩾2λ}⩽1/max⁡(2,λ)\mathbb {P}\left\{ \xi \geqslant 2\lambda \right\} \leqslant 1/\max (2, \lambda ).

Problem 14.9

Let the random variable ξ\xi have:

?
(a)

a Poisson distribution with parameter λ\lambda;

(b)

an exponential distribution with parameter α\alpha;

(c)

the standard normal distribution.

Determine the value of the parameter hh minimizing the expression on the right-hand side of the inequality in problem 14.7. Find the value of this minimum.

Problem 14.10

Prove the following inequalities:

?
(a)

E[ξ4]⩾(E[ξ])4\mathbb {E}\left[\xi^4\right] \geqslant (\mathbb {E}\left[\xi \right])^4;

(b)

E[ξ2]E[ξ4]⩾(E[ξ3])2\mathbb {E}\left[\xi^2\right]\mathbb {E}\left[\xi^4\right] \geqslant (\mathbb {E}\left[\xi^3\right])^2.

When can equality hold in these inequalities?

Problem 14.11

Is the following reasoning valid: «It is 1 km from home to work, and I walk at an average speed of 5 km/h, so on average my commute takes 12 minutes»?

?
Problem 14.12

Prove that if the random variable ξ\xi has a moment of order kk, then its distribution function FF satisfies the relation

lim⁡x→∞xk(1−F(x)+F(−x))=0. \lim _{x \to \infty } x^k (1 - F(x) + F(-x)) = 0.
?
Problem 14.13

Prove that E[ξ2]=0\mathbb {E}\left[\xi^2\right] = 0 is equivalent to P{ξ=0}=1\mathbb {P}\left\{ \xi = 0\right\} = 1.

?
Problem 14.14

Suppose E[∣ξ∣p]=0\mathbb {E}\left[\left|\xi \right|^p\right] = 0 for some p>0p > 0. What can be said about the random variable ξ\xi?

?
Problem 14.15

Let the random variable ξ\xi have finite expectation, with E[ξ  1A]=0\mathbb {E}\left[\xi \; \mathbb {1}_{A}\right] = 0 for every event AA. Prove that P{ξ=0}=1\mathbb {P}\left\{ \xi = 0\right\} = 1.

?
Problem 14.16

Prove that for any nonnegative random variable ξ\xi with finite variance, the inequality

P{ξ=0}⩽Var⁡[ξ]Var⁡[ξ]+(E[ξ])2 \mathbb {P}\left\{ \xi = 0\right\} \leqslant \frac{\operatorname {Var}\left[\xi \right]}{\operatorname {Var}\left[\xi \right] + (\mathbb {E}\left[\xi \right])^2}

holds.

?
Section 15Characteristic Functions[36/50%]
Problem 15.1

Prove that for any random variable ξ\xi (whose characteristic function is defined as ϕξ(t)=E[eitξ]\phi_\xi (t) = \mathbb {E}\left[e^{it\xi }\right], where ii is the imaginary unit):

?
(a)

ϕξ(0)=1\phi_\xi (0) = 1 and ∣ϕξ(t)∣⩽1\left|\phi_\xi (t)\right| \leqslant 1;

(b)

ϕξ(t)‾=ϕξ(−t)=ϕ−ξ(t)\overline{\phi_\xi (t)} = \phi_\xi (-t) = \phi_{-\xi }(t);

(c)

ϕaξ+b(t)=eitbϕξ(at)\phi_{a\xi +b}(t) = e^{itb}\phi_\xi (at) for any constants aa and bb;

(d)

ϕξ+η(t)=ϕξ(t)ϕη(t)\phi_{\xi +\eta }(t) = \phi_\xi (t)\phi_\eta (t), if the random variables ξ\xi and η\eta are independent;

(e)

the characteristic function ϕξ(t)\phi_\xi (t) is uniformly continuous, i.e.,

sup⁡t∈R∣ϕξ(t+h)−ϕξ(t)∣→0 as h→0. \sup _{t \in \mathbb {R}} \left|\phi _\xi (t+h) - \phi _\xi (t)\right| \to 0 \text{ as } h \to 0.
Problem 15.2

Let the random variable ξ\xi have an absolute moment of order kk, i.e., E[∣ξ∣k]<∞\mathbb {E}\left[\left|\xi \right|^k\right] < \infty. Prove that its characteristic function ϕξ(t)\phi_\xi (t):

?
(a)

is continuously differentiable kk times, and its kk-th derivative at zero is expressed in terms of the kk-th moment:

ϕξ(k)(0)=ikE[ξk]; \phi _\xi ^{(k)}(0) = i^k \mathbb {E}\left[\xi ^k\right];
(b)

admits a Taylor expansion in a neighborhood of t=0t = 0:

ϕξ(t)=1+itE[ξ]−t22E[ξ2]+…+iktkk!E[ξk]+o(∣t∣k) \phi _\xi (t) = 1 + it\mathbb {E}\left[\xi \right] - \frac{t^2}{2}\mathbb {E}\left[\xi ^2\right] + \ldots + \frac{i^k t^k}{k!}\mathbb {E}\left[\xi ^k\right] + o(\left|t\right|^k)

as t→0t \to 0.

Problem 15.3

Prove that a characteristic function is even if and only if the distribution of the random variable is symmetric.

?
Problem 15.4

Prove that a characteristic function is real-valued if and only if it is even.

?
Problem 15.5

Compute the characteristic function of the random variable ξ\xi, which has:

?
(a)

a degenerate distribution at the point aa;

(b)

a Bernoulli distribution with parameter pp;

(c)

a binomial distribution with parameters pp and nn;

(d)

a Poisson distribution with parameter λ\lambda;

(e)

a geometric distribution with parameter pp.

Problem 15.6

Compute the characteristic function of the absolutely continuous distribution:

?
(a)

uniform on the segment [0,1][0, 1];

(b)

uniform on the segment [−a,a][-a, a];

(c)

uniform on the segment [a,b][a, b];

(d)

exponential with parameter α\alpha;

(e)

Laplace with parameter α\alpha;

(f)

standard normal;

(g)

normal with parameters aa and σ2\sigma^2;

(h)

Cauchy with location parameter aa and scale parameter 1;

(i)

the Γ\Gamma-distribution with parameters α\alpha and β\beta.

Problem 15.7

Find the distribution law with the following characteristic function:

?
(a)

cos⁡t\cos t;

(b)

∑k=0∞akcos⁡(kt)\displaystyle \sum_{k=0}^\infty a_k \cos (kt), where ak⩾0a_k \geqslant 0 and ∑k=0∞ak=1\displaystyle \sum_{k=0}^\infty a_k = 1;

(c)

cos⁡2t\cos^2 t;

(d)

12+cos⁡t2+isin⁡t6\dfrac {1}{2} + \dfrac {\cos t}{2} + i\dfrac {\sin t}{6};

(e)

e−t2e^{-t^2};

(f)

e−∣t∣e^{-\left|t\right|};

(g)

1/(1+t2)1/(1+t^2);

(h)

1/(1−it)1/(1-it);

(i)

(sin⁡t)/t(\sin t)/t;

(j)

e−∣t∣cos⁡te^{-\left|t\right|}\cos t.

Problem 15.8

Let the probability space be the segment [0,1][0, 1] with the Borel σ\sigma-algebra of subsets and Lebesgue measure. Find the characteristic function of the random variable ξ\xi if:

?
(a)

ξ(ω)=ln⁡ω\xi (\omega ) = \ln \omega, ξ(0)=0\xi (0) = 0;

(b)

ξ(ω)={2ωfor 0⩽ω⩽1/2,2ω−1for 1/2<ω⩽1;\xi (\omega ) = \begin{cases} 2\omega & \text{for } 0 \leqslant \omega \leqslant 1/2, \\ 2\omega - 1 & \text{for } 1/2 < \omega \leqslant 1; \end{cases}

(c)

ξ(ω)={1for 0⩽ω⩽1/3,0for 1/3<ω⩽2/3,1for 2/3<ω⩽1.\xi (\omega ) = \begin{cases} 1 & \text{for } 0 \leqslant \omega \leqslant 1/3, \\ 0 & \text{for } 1/3 < \omega \leqslant 2/3, \\ 1 & \text{for } 2/3 < \omega \leqslant 1. \end{cases}

Problem 15.9

Let ξ\xi and η\eta be independent identically distributed random variables with characteristic function ϕ\phi. Find the characteristic function of the random variable ξ−η\xi - \eta.

?
Problem 15.10

Determine whether the following is a characteristic function: sin⁡t\sin t.

?
Problem 15.11

Determine whether the following is a characteristic function: sin⁡t+1\sin t + 1.

?
Problem 15.12

Determine whether the following is a characteristic function: e−t4e^{-t^4}.

?
Problem 15.13

Determine whether the following is a characteristic function: cos⁡(t2)\cos (t^2).

?
Problem 15.14

Determine whether the following is a characteristic function: cos⁡5t\cos^5 t.

?
Problem 15.15

Determine whether the following is a characteristic function: e−i∣t∣e^{-i\left|t\right|}.

?
Problem 15.16

Determine whether the following is a characteristic function: 11−∣t∣i\dfrac {1}{1 - \left|t\right|i}.

?
Problem 15.17

Determine whether the following is a characteristic function:

ϕ(t)={1−t2,∣t∣<1,0,∣t∣⩾1. \phi (t) = \begin{cases} \sqrt{1-t^2}, & \left|t\right| < 1, \\ 0, & \left|t\right| \geqslant 1. \end{cases}
?
Problem 15.18

Determine whether the following is a characteristic function:

ϕ(t)={1−t2,∣t∣<1,0,∣t∣⩾1. \phi (t) = \begin{cases} 1-t^2, & \left|t\right| < 1, \\ 0, & \left|t\right| \geqslant 1. \end{cases}
?
Problem 15.19

Determine whether the following is a characteristic function:

ϕ(t)={1−∣t∣,∣t∣<1,0,∣t∣⩾1. \phi (t) = \begin{cases} 1-\left|t\right|, & \left|t\right| < 1, \\ 0, & \left|t\right| \geqslant 1. \end{cases}
?
Problem 15.20

Determine whether the following is a characteristic function: a real-valued function that is not even.

?
Problem 15.21

Prove that the equality ϕξ+η(t)=ϕξ(t)ϕη(t)\phi_{\xi +\eta }(t) = \phi_\xi (t)\phi_\eta (t) does not, in general, imply the independence of the random variables ξ\xi and η\eta.

?
Problem 15.22

Let ϕ(t)\phi (t) be an even continuous function, convex for t⩾0t \geqslant 0, such that 0⩽ϕ(t)⩽10 \leqslant \phi (t) \leqslant 1 and ϕ(0)=1\phi (0) = 1. Prove that ϕ(t)\phi (t) is a characteristic function.

?
Problem 15.23

Do there exist distinct characteristic functions that coincide on some segment containing the origin?

?
Problem 15.24

Let the random variable ξ\xi be independent of η\eta and ζ\zeta, with the sums ξ+η\xi + \eta and ξ+ζ\xi + \zeta having the same distribution. Is it true that the random variables η\eta and ζ\zeta have the same distribution?

?
Problem 15.25

Let ϕ1,ϕ2,…\phi_1, \phi_2, \ldots be characteristic functions and a1,a2,…a_1, a_2, \ldots nonnegative numbers such that ∑i=1∞ai=1\displaystyle \sum_{i=1}^\infty a_i = 1. Prove that the function ∑i=1∞aiϕi\displaystyle \sum_{i=1}^\infty a_i \phi_i is also a characteristic function.

?
Problem 15.26

Prove that if ϕ\phi is a characteristic function, then so is the function eϕ−1e^{\phi - 1}.

?
Problem 15.27

Prove that if ϕ\phi is a characteristic function, then so is the function 22−ϕ−1\dfrac {2}{2 - \phi } - 1.

?
Problem 15.28

Prove that if ϕ\phi is a characteristic function, then so is the function ϕ2\phi^2.

?
Problem 15.29

Prove that if ϕ\phi is a characteristic function, then so is the function Re(ϕ)\mathrm{Re}\left(\phi \right).

?
Problem 15.30

Prove that if ϕ\phi is a characteristic function, then so is the function 1t∫0tϕ(z) dz\dfrac {1}{t}\displaystyle \int_0^t \phi (z)\, dz.

?
Problem 15.31

Prove that if ϕ\phi is a characteristic function, then so is the function ∣ϕ∣2\left|\phi \right|^2.

?
Problem 15.32

Let ξ\xi and η\eta be independent random variables with distribution functions FF and GG and characteristic functions ϕ\phi and ψ\psi respectively. Prove that the product ξη\xi \eta has characteristic function

∫−∞∞ϕ(tz) dG(z)=∫−∞∞ψ(tz) dF(z). \int _{-\infty }^\infty \phi (tz)\, dG(z) = \int _{-\infty }^\infty \psi (tz)\, dF(z).
?
Problem 15.33

Let ϕ\phi be a characteristic function. Prove that:

?
(a)

∣ϕ(t+h)−ϕ(t)∣⩽2[1−Re(ϕ(h))]\left|\phi (t+h) - \phi (t)\right| \leqslant \sqrt{2[1 - \mathrm{Re}\left(\phi (h)\right)]};

(b)

1−Re(ϕ(2t))⩽4(1−Re(ϕ(t)))1 - \mathrm{Re}\left(\phi (2t)\right) \leqslant 4(1 - \mathrm{Re}\left(\phi (t)\right)).

Problem 15.34

Prove that for any random variable ξ\xi with characteristic function ϕ\phi, and any x>0x > 0,

P{∣ξ∣>2/x}⩽1x∫−xx(1−ϕ(u)) du. \mathbb {P}\left\{ \left|\xi \right| > 2/x\right\} \leqslant \frac{1}{x}\int _{-x}^x (1 - \phi (u))\, du.
?
Problem 15.35

Consider a distribution symmetric about 0. Let F(x)F(x) be the corresponding distribution function and ϕ(t)\phi (t) the characteristic function. Prove that if

x(1−F(x))→c>0 as x→∞, x(1 - F(x)) \to c > 0 \text{ as } x \to \infty ,

then

ϕ(t)=1−cπ∣t∣+o(t) as t→0. \phi (t) = 1 - c\pi \left|t\right| + o(t) \text{ as } t \to 0.
?
Problem 15.36

Prove that if the quantities ξ\xi and η\eta are independent, identically distributed, and their sum is normally distributed, then ξ\xi and η\eta are also normally distributed.

?
Section 16Generating Functions[13/54%]
Problem 16.1

Let the random variable ξ\xi take only nonnegative integer values, and let ψ(z)\psi (z), ∣z∣⩽1\left|z\right| \leqslant 1, be its generating function (that is, ψ(z)=∑n=0∞znP{ξ=n}=E[zξ]\psi (z) = \sum_{n=0}^\infty z^n \mathbb {P}\left\{ \xi =n\right\} = \mathbb {E}\left[z^\xi \right]). Does the generating function ψ(z)\psi (z) uniquely determine the distribution of the random variable ξ\xi?

?
Problem 16.2

For any two independent random variables ξ\xi and η\eta, prove that ψξ+η(z)=ψξ(z)ψη(z)\psi_{\xi +\eta }(z) = \psi_\xi (z)\psi_\eta (z).

?
Problem 16.3

Let ξ\xi be a nonnegative integer-valued random variable with generating function ψ(z)\psi (z). Find the generating functions of the quantities:

?
(a)

ξ+1\xi + 1;

(b)

2ξ2\xi;

(c)

3ξ+53\xi + 5.

Problem 16.4

Find the generating functions of random variables having:

?
(a)

a Bernoulli distribution with parameter pp;

(b)

a binomial distribution with parameters pp and nn;

(c)

a Poisson distribution with parameter λ\lambda;

(d)

a geometric distribution with parameter pp.

Problem 16.5

Let ξ\xi be a nonnegative integer-valued random variable with generating function ψξ\psi_\xi. In terms of ψξ\psi_\xi, express the generating functions of the following sequences:

?
(a)

P{ξ⩽n}\mathbb {P}\left\{ \xi \leqslant n\right\};

(b)

P{ξ⩾n}\mathbb {P}\left\{ \xi \geqslant n\right\};

(c)

P{ξ<n}\mathbb {P}\left\{ \xi < n\right\};

(d)

P{ξ>n}\mathbb {P}\left\{ \xi > n\right\}.

Problem 16.6

Let {ξn}\left\{ \xi_n\right\} be a sequence of nonnegative integer-valued independent identically distributed random variables with generating function ψξ\psi_\xi. Let ν\nu be a positive integer-valued random variable, independent of {ξn}\left\{ \xi_n\right\}, with generating function ψν\psi_\nu. Find the generating function of the sum of a random number of terms ξ1+…+ξν\xi_1 + \ldots + \xi_\nu.

?
Problem 16.7

Find the distribution laws corresponding to the following generating functions:

?
(a)

(1+z)24\dfrac {(1+z)^2}{4};

(b)

e5(z−1)e^{5(z-1)};

(c)

12(1−z/2)\dfrac {1}{2(1 - z/2)};

(d)

(1/3+2z/3)n(1/3 + 2z/3)^n, n∈Nn \in \mathbb {N}.

Problem 16.8

Let the random variable ξ\xi have generating function ψ\psi. Find the distribution of ξ\xi, if ψ(1/2n)=1/2n\psi (1/2^n) = 1/2^n for every nn.

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Problem 16.9

Prove that the function ψ(z)=∣z∣\psi (z) = \left|z\right| cannot be the generating function of a probability distribution.

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Problem 16.10

Let the random variable ξ\xi take the values 0,1,…,n−10, 1, \ldots , n-1 with probability 1/n1/n each. Prove that if nn is composite, then ξ\xi can be represented as the sum of independent integer-valued random variables.

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Problem 16.11

Let the random variable ξ\xi take the values 0 and 1 with probability 1/21/2 each, and let η\eta take the values 0, 1, 2, and 3 with probabilities 1/81/8, 1/41/4, 1/21/2, and 1/81/8 respectively. Prove that there does not exist a random variable ζ\zeta, independent of ξ\xi, such that ξ+ζ=η\xi + \zeta = \eta.

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Problem 16.12

Solve problem 10.39 using generating functions.

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Problem 16.13

Let {ξn}\left\{ \xi_n\right\} be a sequence of independent identically distributed random variables with common distribution function FF. Let ν\nu be a positive integer-valued random variable, independent of {ξn}\left\{ \xi_n\right\}, with generating function ψ\psi. Prove that the distribution function of the random variable max⁡(ξ1,…,ξν)\max (\xi_1, \ldots , \xi_\nu ) equals ψ(F(x))\psi (F(x)).

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Section 17Infinitely Divisible Distributions[13/0%]
Problem 17.1

Let the random variable ξ\xi have an infinitely divisible distribution (that is, for every natural nn there exist independent identically distributed random variables ξ1,…,ξn\xi_1, \ldots , \xi_n such that the distribution of the sum ξ1+…+ξn\xi_1 + \ldots + \xi_n coincides with the distribution of ξ\xi). Prove that for any real aa and bb, the distribution of the random variable aξ+ba\xi + b is also infinitely divisible.

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Problem 17.2

Prove that a random variable ξ\xi with characteristic function ϕ(t)\phi (t) has an infinitely divisible distribution if and only if, for every n>0n > 0, the function ϕ1/n(t)\phi^{1/n}(t) is the characteristic function of some distribution.

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Problem 17.3

Let ϕ(t)\phi (t) be the characteristic function of an infinitely divisible distribution. Prove that for any c>0c > 0 the function ϕc(t)\phi^c(t) is also the characteristic function of an infinitely divisible distribution.

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Problem 17.4

Prove that the characteristic function of an infinitely divisible distribution is nowhere zero.

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Problem 17.5

Let ξ\xi and η\eta be independent random variables with infinitely divisible distributions. Prove that the distribution of the random variable ξ+η\xi + \eta is also infinitely divisible.

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Problem 17.6

Let ϕ(t)\phi (t) be the characteristic function of an infinitely divisible distribution. Prove that the function ∣ϕ(t)∣\left|\phi (t)\right| is also the characteristic function of an infinitely divisible distribution.

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Problem 17.7

Prove the infinite divisibility of:

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(a)

the Poisson distribution;

(b)

the normal distribution;

(c)

the Cauchy distribution;

(d)

the Γ\Gamma-distribution;

(e)

the exponential distribution;

(f)

the chi-squared distribution.

Problem 17.8

Let ϕ(t)\phi (t) be a characteristic function. Prove that for any λ>0\lambda > 0 the function eλ(ϕ(t)−1)e^{\lambda (\phi (t)-1)} is the characteristic function of an infinitely divisible distribution.

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Problem 17.9

Prove that if the sum of two independent infinitely divisible random variables has the Poisson distribution, then each of the summands also has the Poisson distribution.

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Problem 17.10

Prove that if the sum of two independent infinitely divisible random variables has the normal distribution, then each of the summands also has the normal distribution.

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Problem 17.11

Prove that the uniform distribution on a segment cannot be infinitely divisible.

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Problem 17.12

Prove that no non-degenerate infinitely divisible distribution can be concentrated on a finite segment.

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Problem 17.13

Let ξ\xi and η\eta be independent random variables, with ξ\xi having a uniform distribution on some segment. Prove that the random variable ξ+η\xi + \eta cannot have an infinitely divisible distribution.

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