Math Problems Vault

Problem 2.1

Let $X$ and $Y$ be independent $\text{Exp}(1)$ -distributed random variables. Find the conditional distribution of $X$ given that $X+Y=c$ ( $c$ is a positive constant).

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Problem 2.2

Let $X$ and $Y$ be independent $\Gamma (2,a)$ -distributed random variables. Find the conditional distribution of $X$ given that $X+Y=2$ .

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Problem 2.3

(a)(b)

The life of a repairing device is $\text{Exp}(1/a)$ -distributed. Peter wishes to use it on $n$ different, independent, $\text{Exp}(1/na)$ -distributed occasions.

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(a)

Compute the probability $P_n$ that this is possible.

(b)

Determine the limit of $P_n$ as $n \rightarrow \infty$ .

Problem 2.4

The life $T$ (hours) of the lightbulb in an overhead projector follows an $\text{Exp}(10)$ -distribution. During a normal week it is used a $\text{Po}(12)$ -distributed number of lectures lasting exactly one hour each. Find the probability that a projector with a newly installed lightbulb functions throughout a normal week (without replacing the lightbulb).

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Problem 2.5

The random variables $N, X_1, X_2, \ldots$ are independent, $N \in \text{Po}(\lambda )$ , and $X_k \in \text{Be}(1/2)$ , $k \geq 1$ . Set

Y_1 = \sum _{k=1}^{N} X_k \quad \text{and} \quad Y_2 = N - Y_1

( $Y_1 = 0$ for $N=0$ ). Show that $Y_1$ and $Y_2$ are independent, and determine their distributions.

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Problem 2.6

Suppose that $X \in N(0,1)$ and $Y \in \text{Exp}(1)$ are independent random variables. Prove that $X\sqrt{2Y}$ has a standard Laplace distribution.

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Problem 2.7

(a)(b)

Let $N \in \text{Ge}(p)$ and set $X = (-1)^N$ . Compute

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(a)

$\mathbb {E}\left[X\right]$ and $\operatorname {Var}\left[X\right]$ ,

(b)

the distribution (probability function) of $X$ .

Problem 2.8

(a)(b)(c)

The density function of the two-dimensional random variable $(X,Y)$ is

f_{X,Y}(x,y) = \begin{cases} \dfrac {x^2}{2 \cdot y^3} \cdot e^{-\frac{x}{y}}, & \text{for} \quad 0 < x < \infty , \quad 0 < y < 1, \\ 0, & \text{otherwise}. \end{cases}

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(a)

Determine the distribution of $Y$ .

(b)

Find the conditional distribution of $X$ given that $Y=y$ .

(c)

Use the results from (a) and (b) to compute $\mathbb {E}\left[X\right]$ and $\operatorname {Var}\left[X\right]$ .

Problem 2.9

(a)(b)

The density of the random vector $(X,Y)'$ is

f_{X,Y}(x,y) = \begin{cases} cx, & \text{for} \quad x \geq 0, \quad y \geq 0, \quad x+y \leq 1, \\ 0, & \text{otherwise}. \end{cases}

Compute

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(a)

$c$ ,

(b)

the conditional expectations $\mathbb {E}\left[ Y \mid X=x \right]$ and $\mathbb {E}\left[ X \mid Y=y \right]$ .

Problem 2.10

Suppose $X$ and $Y$ have a joint density function given by

f(x,y) = \begin{cases} cx^2, & \text{for} \quad 0 < x < y < 1, \\ 0, & \text{otherwise}. \end{cases}

Find $c$ , the marginal density functions, $\mathbb {E}\left[X\right]$ , $\mathbb {E}\left[Y\right]$ , and the conditional expectations $\mathbb {E}\left[ Y \mid X=x \right]$ and $\mathbb {E}\left[ X \mid Y=y \right]$ .

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Problem 2.11

Suppose $X$ and $Y$ have a joint density function given by

f(x,y) = \begin{cases} c \cdot x^2 y, & \text{for} \quad 0 < y < x < 1, \\ 0, & \text{otherwise}. \end{cases}

Compute $c$ , the marginal densities, $\mathbb {E}\left[X\right]$ , $\mathbb {E}\left[Y\right]$ , and the conditional expectations $\mathbb {E}\left[ Y \mid X=x \right]$ and $\mathbb {E}\left[ X \mid Y=y \right]$ .

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Problem 2.12

Let $X$ and $Y$ have joint density

f(x,y) = \begin{cases} cxy, & \text{when} \quad 0 < y < x < 1, \\ 0, & \text{otherwise}. \end{cases}

Compute the conditional expectations $\mathbb {E}\left[ Y \mid X=x \right]$ and $\mathbb {E}\left[ X \mid Y=y \right]$ .

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Problem 2.13

Let $X$ and $Y$ have joint density

f(x,y) = \begin{cases} cy, & \text{when} \quad 0 < y < x < 2, \\ 0, & \text{otherwise}. \end{cases}

Compute the conditional expectations $\mathbb {E}\left[ Y \mid X=x \right]$ and $\mathbb {E}\left[ X \mid Y=y \right]$ .

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Problem 2.14

Suppose that $X$ and $Y$ are random variables with joint density

f(x,y) = \begin{cases} c(x+2y), & \text{when} \quad 0 < x < y < 1, \\ 0, & \text{otherwise}. \end{cases}

Compute the regression functions $\mathbb {E}\left[ Y \mid X=x \right]$ and $\mathbb {E}\left[ X \mid Y=y \right]$ .

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Problem 2.15

Suppose that $X$ and $Y$ are random variables with a joint density

f(x,y) = \begin{cases} \dfrac {2}{5}(2x+3y), & \text{when} \quad 0 < x,y < 1, \\ 0, & \text{otherwise}. \end{cases}

Compute the conditional expectations $\mathbb {E}\left[ Y \mid X=x \right]$ and $\mathbb {E}\left[ X \mid Y=y \right]$ .

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Problem 2.16

Let $X$ and $Y$ be random variables with a joint density

f(x,y) = \begin{cases} \dfrac {4}{5}(x+3y)e^{-x-2y}, & \text{when} \quad x,y > 0, \\ 0, & \text{otherwise}. \end{cases}

Compute the regression functions $\mathbb {E}\left[ Y \mid X=x \right]$ and $\mathbb {E}\left[ X \mid Y=y \right]$ .

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Problem 2.17

Suppose that the joint density of $X$ and $Y$ is given by

f(x,y) = \begin{cases} xe^{-x-xy}, & \text{when} \quad x > 0, \quad y > 0, \\ 0, & \text{otherwise}. \end{cases}

Determine the regression functions $\mathbb {E}\left[ Y \mid X=x \right]$ and $\mathbb {E}\left[ X \mid Y=y \right]$ .

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Problem 2.18

Let the joint density function of $X$ and $Y$ be given by

f(x,y) = \begin{cases} c(x+y), & \text{for} \quad 0 < x < y < 1, \\ 0, & \text{otherwise}. \end{cases}

Determine $c$ , the marginal densities, $\mathbb {E}\left[X\right]$ , $\mathbb {E}\left[Y\right]$ , and the conditional expectations $\mathbb {E}\left[ Y \mid X=x \right]$ and $\mathbb {E}\left[ X \mid Y=y \right]$ .

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Problem 2.19

Let the joint density of $X$ and $Y$ be given by

f_{X,Y}(x,y) = \begin{cases} c, & \text{for} \quad 0 \leq x \leq 1, \quad x^2 \leq y \leq x, \\ 0, & \text{otherwise}. \end{cases}

Compute $c$ , the marginal densities, and the conditional expectations $\mathbb {E}\left[ Y \mid X=x \right]$ and $\mathbb {E}\left[ X \mid Y=y \right]$ .

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Problem 2.20

Suppose that $X$ and $Y$ are random variables with joint density

f(x,y) = \begin{cases} cx, & \text{when} \quad 0 < x < 1, \quad x^3 < y < x^{1/3}, \\ 0, & \text{otherwise}. \end{cases}

Compute the conditional expectations $\mathbb {E}\left[ Y \mid X=x \right]$ and $\mathbb {E}\left[ X \mid Y=y \right]$ .

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Problem 2.21

Suppose that $X$ and $Y$ are random variables with joint density

f(x,y) = \begin{cases} cy, & \text{when} \quad 0 < x < 1, \quad x^4 < y < x^{1/4}, \\ 0, & \text{otherwise}. \end{cases}

Compute the conditional expectations $\mathbb {E}\left[ Y \mid X=x \right]$ and $\mathbb {E}\left[ X \mid Y=y \right]$ .

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Problem 2.22

Let the joint density function of $X$ and $Y$ be given by

f(x,y) = \begin{cases} c \cdot x^3 y, & \text{for} \quad x,y > 0, \quad x^2+y^2 \leq 1, \\ 0, & \text{otherwise}. \end{cases}

Compute $c$ , the marginal densities, and the conditional expectations $\mathbb {E}\left[ Y \mid X=x \right]$ and $\mathbb {E}\left[ X \mid Y=y \right]$ .

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Problem 2.23

The joint density function of $X$ and $Y$ is given by

f(x,y) = \begin{cases} c \cdot xy, & \text{for} \quad x,y > 0, \quad 4x^2+y^2 \leq 1, \\ 0, & \text{otherwise}. \end{cases}

Compute $c$ , the marginal densities, and the conditional expectations $\mathbb {E}\left[ Y \mid X=x \right]$ and $\mathbb {E}\left[ X \mid Y=y \right]$ .

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Problem 2.24

Let $X$ and $Y$ have joint density

f(x,y) = \begin{cases} \dfrac {c}{x^3 y}, & \text{when} \quad 1 < y < x, \\ 0, & \text{otherwise}. \end{cases}

Compute the conditional expectations $\mathbb {E}\left[ Y \mid X=x \right]$ and $\mathbb {E}\left[ X \mid Y=y \right]$ .

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Problem 2.25

Let $X$ and $Y$ have joint density

f(x,y) = \begin{cases} \dfrac {c}{x^4 y}, & \text{when} \quad 1 < y < x, \\ 0, & \text{otherwise}. \end{cases}

Compute the conditional expectations $\mathbb {E}\left[ Y \mid X=x \right]$ and $\mathbb {E}\left[ X \mid Y=y \right]$ .

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Problem 2.26

Suppose that $X$ and $Y$ are random variables with a joint density

f(x,y) = \begin{cases} \dfrac {c}{(1+x-y)^2}, & \text{when} \quad 0 < y < x < 1, \\ 0, & \text{otherwise}. \end{cases}

Compute the conditional expectations $\mathbb {E}\left[ Y \mid X=x \right]$ and $\mathbb {E}\left[ X \mid Y=y \right]$ .

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Problem 2.27

Suppose that $X$ and $Y$ are random variables with a joint density

f(x,y) = \begin{cases} c \cdot \cos x, & \text{when} \quad 0 < y < x < \frac{\pi }{2}, \\ 0, & \text{otherwise}. \end{cases}

Compute the conditional expectations $\mathbb {E}\left[ Y \mid X=x \right]$ and $\mathbb {E}\left[ X \mid Y=y \right]$ .

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Problem 2.28

Let $X$ and $Y$ have joint density

f(x,y) = \begin{cases} c\log y, & \text{when} \quad 0 < y < x < 1, \\ 0, & \text{otherwise}. \end{cases}

Compute the conditional expectations $\mathbb {E}\left[ Y \mid X=x \right]$ and $\mathbb {E}\left[ X \mid Y=y \right]$ .

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Problem 2.29

The random vector $(X,Y)'$ has the following joint distribution:

\mathbb {P}\left(X=m, Y=n\right) = \binom {m}{n}\frac{1}{2^m}\frac{m}{15},

where $m=1,2,\ldots ,5$ and $n=0,1,\ldots ,m$ . Compute $\mathbb {E}\left[ Y \mid X=m \right]$ .

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Problem 2.30

Show that a suitable power of a Weibull-distributed random variable whose parameter is gamma-distributed is Pareto-distributed. More precisely, show that if

X \mid A=a \in W\left(\frac{1}{a}, \frac{1}{b}\right) \quad \text{with} \quad A \in \Gamma (p,\theta ),

then $X^b$ has a (translated) Pareto distribution.

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Problem 2.31

Show that an exponential random variable such that the inverse of the parameter is gamma-distributed is Pareto-distributed. More precisely, show that if

X \mid M=m \in \text{Exp}(m) \quad \text{with} \quad M^{-1} \in \Gamma (p,a),

then $X$ has a (translated) Pareto distribution.

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Problem 2.32

(a)(b)(c)

Let $X$ and $Y$ be random variables such that

Y \mid X=x \in \text{Exp}(1/x) \quad \text{with} \quad X \in \Gamma (2,1).

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(a)

Show that $Y$ has a translated Pareto distribution.

(b)

Compute $\mathbb {E}\left[Y\right]$ .

(c)

Check the value in (b) by recomputing it via our favorite formula for conditional means.

Problem 2.33

Suppose that the random variable $X$ is uniformly distributed symmetrically around zero, but in such a way that the parameter is uniform on $(0,1)$ ; that is, suppose that

X \mid A=a \in U(-a,a) \quad \text{with} \quad A \in U(0,1).

Find the distribution of $X$ , $\mathbb {E}\left[X\right]$ , and $\operatorname {Var}\left[X\right]$ .

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Problem 2.34

(a)(b)

In Section 4 we studied the situation when a coin, such that $p = \mathbb {P}\left(\text{head}\right)$ is considered to be a $U(0,1)$ -distributed random variable, is tossed, and found that if $X_n = \#$ heads after $n$ tosses, then $X_n$ is uniformly distributed over the integers $0,1,\ldots ,n$ .

Suppose instead that $p$ is considered to be $\beta (2,2)$ -distributed. What then? More precisely, consider the following model:

X_n \mid Y=y \in \text{Bin}(n,y) \quad \text{with} \quad f_Y(y) = 6y(1-y), \quad 0 < y < 1.

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(a)

Compute $\mathbb {E}\left[X_n\right]$ and $\operatorname {Var}\left[X_n\right]$ .

(b)

Determine the distribution of $X_n$ .

Problem 2.35

Let $X$ and $Y$ be jointly distributed random variables such that

Y \mid X=x \in \text{Bin}(n,x) \quad \text{with} \quad X \in U(0,1).

Compute $\mathbb {E}\left[Y\right]$ , $\operatorname {Var}\left[Y\right]$ , and $\operatorname {Cov}\left[ X, Y \right]$ (without using what is known from Section 4 about the distribution of $Y$ ).

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Problem 2.36

Let $X$ and $Y$ be jointly distributed random variables such that

Y \mid X=x \in \text{Fs}(x) \quad \text{with} \quad f_X(x) = 3x^2, \quad 0 \leq x \leq 1.

Compute $\mathbb {E}\left[Y\right]$ , $\operatorname {Var}\left[Y\right]$ , $\operatorname {Cov}\left[ X, Y \right]$ , and the distribution of $Y$ .

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Problem 2.37

(a)(b)(c)

Let $X$ be the number of coin tosses until heads is obtained. Suppose that the probability of heads is unknown in the sense that we consider it to be a random variable $Y \in U(0,1)$ .

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(a)

Find the distribution of $X$ (cf. problem 3.8.48).

(b)

The expected value of an $\text{Fs}$ -distributed random variable exists, as is well known. What about $\mathbb {E}\left[X\right]$ ?

(c)

Suppose that the value $X=n$ has been observed. Find the posterior distribution of $Y$ , that is, the distribution of $Y \mid X=n$ .

Problem 2.38

Let $p$ be the probability that the tip points downward after a person throws a drawing pin once. Annika throws a drawing pin until it points downward for the first time. Let $X$ be the number of throws for this to happen. She then throws the drawing pin another $X$ times. Let $Y$ be the number of times the drawing pin points downward in the latter series of throws. Find the distribution of $Y$ (cf. problem 3.8.31).

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Problem 2.39

A point $P$ is chosen uniformly in an $n$ -dimensional sphere of radius $1$ . Next, a point $Q$ is chosen uniformly within the concentric sphere, centered at the origin, going through $P$ . Let $X$ and $Y$ be the distances of $P$ and $Q$ , respectively, to the common center. Find the joint density function of $X$ and $Y$ and the conditional expectations $\mathbb {E}\left[ Y \mid X=x \right]$ and $\mathbb {E}\left[ X \mid Y=y \right]$ .

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Remark.

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Hint 1. Begin by trying the case $n=2$ .

Hint 2. The volume of an $n$ -dimensional sphere of radius $r$ is equal to $c_n r^n$ , where $c_n$ is some constant (which is of no interest for the problem).

Remark. For $n=1$ we rediscover the stick from Example 2.1.

Problem 2.40

Theorem 2.2(b): if $X$ and $Y$ are independent random variables, then $\mathbb {E}\left[ Y \mid X \right] = \mathbb {E}\left[Y\right]$ . Remark 2.4 explains that this holds because the conditional distribution and the unconditional distribution of $Y$ coincide when $X$ and $Y$ are independent, and hence so do the conditional and unconditional expectations.

Let $X$ and $Y$ be independent random variables. The conditional distribution of $Y$ given that $X=x$ then does not depend on $x$ . Moreover, $\mathbb {E}\left[ Y \mid X=x \right]$ is independent of $x$ ; recall Theorem 2.2(b) and Remark 2.4.

Now, suppose instead that $\mathbb {E}\left[ Y \mid X=x \right]$ is independent of $x$ (i.e., that $\mathbb {E}\left[ Y \mid X \right] = \mathbb {E}\left[Y\right]$ ). We say that $Y$ has constant regression with respect to $X$ . However, it does not necessarily follow that $X$ and $Y$ are independent. Namely, let the joint density of $X$ and $Y$ be given by

f(x,y) = \begin{cases} \dfrac {1}{2}, & \text{for} \quad \left|x\right|+\left|y\right| \leq 1, \\ 0, & \text{otherwise}. \end{cases}

Show that $Y$ has constant regression with respect to $X$ and/but that $X$ and $Y$ are not independent.

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