Math Problems Vault

Problem 2.7.1

Each toss of a coin results in a head with probability $p$ . The coin is tossed until the first head appears. Let $X$ be the total number of tosses. What is $\mathbb {P}\left(X > m\right)$ ? Find the distribution function of the random variable $X$ .

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Problem 2.7.2

(a)(b)(c)

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(a)

Show that any discrete random variable may be written as a linear combination of indicator variables.

(b)

Show that any random variable may be expressed as the limit of an increasing sequence of discrete random variables.

(c)

Show that the limit of any increasing convergent sequence of random variables is a random variable.

Problem 2.7.3

(a)(b)

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(a)

Show that, if $X$ and $Y$ are random variables on a probability space $(\Omega , \mathcal{F}, \mathbb {P})$ , then so are $X+Y, X Y$ , and $\min \left\{ X, Y\right\}$ .

(b)

Show that the set of all random variables on a given probability space $(\Omega , \mathcal{F}, \mathbb {P})$ constitutes a vector space over the reals. If $\Omega$ is finite, write down a basis for this space.

Problem 2.7.4

Let $X$ have distribution function

F(x) = \begin{cases} 0 & \text{ if } x < 0 \\ \frac{1}{2} x & \text{ if } 0 \leq x \leq 2 \\ 1 & \text{ if } x > 2\end{cases}

and let $Y = X^{2}$ . Find

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(a)

$\mathbb {P}\left(\frac{1}{2} \leq X \leq \frac{3}{2}\right)$ ,

(b)

$\mathbb {P}\left(1 \leq X < 2\right)$ ,

(c)

$\mathbb {P}\left(Y \leq X\right)$ ,

(d)

$\mathbb {P}\left(X \leq 2 Y\right)$ ,

(e)

$\mathbb {P}\left(X+Y \leq \frac{3}{4}\right)$

(f)

the distribution function of $Z = \sqrt{X}$ .

Problem 2.7.5

(a)(b)(c)

Let $X$ have distribution function

F(x) = \begin{cases} 0 & \text{ if } x < -1 \\ 1-p & \text{ if }-1 \leq x < 0 \\ 1-p+\frac{1}{2} x p & \text{ if } 0 \leq x \leq 2 \\ 1 & \text{ if } x > 2\end{cases}

Sketch this function, and find:

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(a)

$\mathbb {P}\left(X = -1\right)$ ,

(b)

$\mathbb {P}\left(X = 0\right)$ ,

(c)

$\mathbb {P}\left(X \geq 1\right)$ .

Problem 2.7.6

Buses arrive at ten minute intervals starting at noon. A man arrives at the bus stop a random number $X$ minutes after noon, where $X$ has distribution function

\mathbb {P}\left(X \leq x\right) = \begin{cases} 0 & \text{ if } x < 0 \\ x / 60 & \text{ if } 0 \leq x \leq 60 \\ 1 & \text{ if } x > 60\end{cases}

What is the probability that he waits less than five minutes for a bus?

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Problem 2.7.7

Airlines find that each passenger who reserves a seat fails to turn up with probability $\frac{1}{10}$ independently of the other passengers. EasyPeasy Airlines always sell 10 tickets for their 9 seat aeroplane while RyeLoaf Airways always sell 20 tickets for their 18 seat aeroplane. Which is more often over-booked?

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Problem 2.7.8

A fairground performer claims the power of telekinesis. The crowd throws coins and he wills them to fall heads up. He succeeds five times out of six. What chance would he have of doing at least as well if he had no supernatural powers?

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Problem 2.7.9

(a)(b)(c)(d)

Express the distribution functions of

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(a)

$X^{+} = \max \left\{ 0, X\right\}$

(b)

$X^{-} = -\min \left\{ 0, X\right\}$

(c)

$\left|X\right| = X^{+}+X^{-}$

(d)

$-X$

in terms of the distribution function $F$ of the random variable $X$ .

Problem 2.7.10

Show that $F_{X}(x)$ is continuous at $x = x_{0}$ if and only if $\mathbb {P}\left(X = x_{0}\right) = 0$ .

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Problem 2.7.11

(a)(b)

The real number $m$ is called a median of the distribution function $F$ whenever $\lim_{y \uparrow m} F(y) \leq \frac{1}{2} \leq F(m)$ .

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(a)

Show that every distribution function $F$ has at least one median, and that the set of medians of $F$ is a closed interval of $\mathbb {R}$ .

(b)

Show, if $F$ is continuous, that $F(m) = \frac{1}{2}$ for any median $m$ .

Problem 2.7.12

(a)(b)

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(a)

Show that it is not possible to weight two dice in such a way that the sum of the two numbers shown by these loaded dice is equally likely to take any value between 2 and 12 (inclusive).

(b)

Given a fair die and a loaded die, show that the sum of their scores, modulo $6$ , has the same distribution as a fair die, irrespective of the loading.

Problem 2.7.13

(a)(b)

A function $d: S \times S \rightarrow \mathbb {R}$ is called a metric on $S$ if:

(i) $d(s, t) = d(t, s) \geq 0$ for all $s, t \in S$ ,

(ii) $d(s, t) = 0$ if and only if $s = t$ , and

(iii) $d(s, t) \leq d(s, u)+d(u, t)$ for all $s, t, u \in S$ .

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(a)

Lévy metric. Let $F$ and $G$ be distribution functions and define the Lévy metric

d_{\mathrm{L}}(F, G) = \inf \left\{ \epsilon > 0: G(x-\epsilon )-\epsilon \leq F(x) \leq G(x+\epsilon )+\epsilon \text{ for all } x\right\} .

Show that $d_{\mathrm{L}}$ is indeed a metric on the space of distribution functions.

(b)

Total variation distance. Let $X$ and $Y$ be integer-valued random variables, and let

d_{\mathrm{TV}}(X, Y) = \sum _{k}|\mathbb {P}\left(X = k\right)-\mathbb {P}\left(Y = k\right)|

Show that $d_{\text{TV }}$ satisfies (i) and (iii) with $S$ the space of integer-valued random variables, and that $d_{\mathrm{TV}}(X, Y) = 0$ if and only if $X$ and $Y$ have the same distribution. Thus $d_{\mathrm{TV}}$ is a metric on the space of equivalence classes of $S$ with equivalence relation given by $X \sim Y$ if $X$ and $Y$ have the same distribution. We call $d_{\mathrm{TV}}$ the total variation distance.

Show that

d_{\mathrm{TV}}(X, Y) = 2 \sup _{A \subseteq \mathbb {Z}}|\mathbb {P}\left(X \in A\right)-\mathbb {P}\left(Y \in A\right)|

Problem 2.7.14

(a)(b)

Ascertain in the following cases whether or not $F$ is the joint distribution function of some pair $(X, Y)$ of random variables. If your conclusion is affirmative, find the distribution functions of $X$ and $Y$ separately.

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(a)

F(x, y) = \begin{cases} 1-e^{-x-y} & \text{ if } x, y \geq 0 \\ 0 & \text{ otherwise } \end{cases}

(b)

F(x, y) = \begin{cases} 1-e^{-x}-x e^{-y} & \text{ if } 0 \leq x \leq y \\ 1-e^{-y}-y e^{-y} & \text{ if } 0 \leq y \leq x \\ 0 & \text{ otherwise } \end{cases}

Problem 2.7.15

It is required to place in order $n$ books $B_{1}, B_{2}, \ldots , B_{n}$ on a library shelf in such a way that readers searching from left to right waste as little time as possible on average. Assuming that each reader requires book $B_{i}$ with probability $p_{i}$ , find the ordering of the books which minimizes $\mathbb {P}\left(T \geq k\right)$ for all $k$ , where $T$ is the (random) number of titles examined by a reader before discovery of the required book.

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Problem 2.7.16

Three coins each show heads with probability $\frac{3}{5}$ and tails otherwise. The first counts 10 points for a head and 2 for a tail, the second counts 4 points for both head and tail, and the third counts 3 points for a head and 20 for a tail.

You and your opponent each choose a coin; you cannot choose the same coin. Each of you tosses your coin and the person with the larger score wins $£ 10^{10}$ . Would you prefer to be the first to pick a coin or the second?

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Problem 2.7.17

Before the development of radar, inertial navigation, and GPS, flying to isolated islands (for example, from Los Angeles to Hawaii) was somewhat 'hit or miss'. In heavy cloud or at night it was necessary to fly by dead reckoning, and then to search the surface. With the aid of a radio, the pilot had a good idea of the correct great circle along which to search, but could not be sure which of the two directions along this great circle was correct (since a strong tailwind could have carried the plane over its target). When you are the pilot, you calculate that you can make $n$ searches before your plane will run out of fuel. On each search you will discover the island with probability $p$ (if it is indeed in the direction of the search) independently of the results of other searches; you estimate initially that there is probability $\alpha$ that the island is ahead of you. What policy should you adopt in deciding the directions of your various searches in order to maximize the probability of locating the island?

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Problem 2.7.18

(a)(b)

Eight pawns are placed randomly on a chessboard, no more than one to a square. What is the probability that:

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(a)

they are in a straight line (do not forget the diagonals)?

(b)

no two are in the same row or column?

Problem 2.7.19

(a)(b)(c)(d)

Which of the following are distribution functions? For those that are, give the corresponding density function $f$ .

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(a)

$F(x) = \begin{cases} 1-e^{-x^{2}} & x \geq 0, \\ 0 & \text{ otherwise. }\end{cases}$

(b)

$F(x) = \begin{cases} e^{-1 / x} & x > 0, \\ 0 & \text{ otherwise. }\end{cases}$

(c)

$F(x) = e^{x} /\left(e^{x}+e^{-x}\right), x \in \mathbb {R}$ .

(d)

$F(x) = e^{-x^{2}}+e^{x} /\left(e^{x}+e^{-x}\right), x \in \mathbb {R}$

Problem 2.7.20

(a)(b)

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(a)

If $U$ and $V$ are jointly continuous, show that $\mathbb {P}\left(U = V\right) = 0$ .

(b)

Let $X$ be uniformly distributed on $(0,1)$ , and let $Y = X$ . Then $X$ and $Y$ are continuous, and $\mathbb {P}\left(X = Y\right) = 1$ . Is there a contradiction here?

Problem 2.7.21

Let $X$ be uniformly distributed on the interval $[0,1]$ , and express $X$ as a continued fraction thus:

X = \cfrac {1}{Y_{1}+\cfrac {1}{Y_{2}+\cfrac {1}{Y_{3}+\ddots }}}

Show that the joint mass function of $Y_{1}$ and $Y_{2}$ is

f(u, v) = \frac{1}{(u v+1)(u v+u+1)}, \quad u, v = 1,2, \ldots

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Problem 2.7.22

(a)(b)

Let $V$ be a vector space of dimension $n$ over a finite field $\mathbb {F}$ with $q$ elements. Let $X_{1}, X_{2}, \ldots , X_{m}$ be independent random variables, each uniformly distributed on $V$ .

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(a)

Let $a_{i} \in \mathbb {F}, i = 1,2, \ldots , m$ , be not all zero. Show that the linear combination $Z = \sum_{i} a_{i} X_{i}$ is uniformly distributed on $V$ .

(b)

Let $p_{m}$ be the probability that $X_{1}, X_{2}, \ldots , X_{m}$ are linearly dependent. Show that, if $m \leq n+1$ ,

q^{-(n-m-1)} \leq p_{m} \leq q^{-(n-m)}, \quad m = 1,2, \ldots , n+1

Problem 2.7.23

(a)(b)(c)

A random variable $X$ with distribution function $F$ is said to be unimodal about a mode $M$ if $F$ is convex on $(-\infty , M)$ and concave on $(M, \infty )$ . Show that, if $F$ is unimodal about $M$ , then the following hold.

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(a)

$F$ is absolutely continuous, except possibly for an atom at $M$ .

(b)

If $F$ is differentiable, then it has a density that is non-decreasing on $(-\infty , M)$ and non-increasing on $(M, \infty )$ , and furthermore, the set of modes of $F$ is a closed bounded interval. [Cf. Problem (2.7.11).]

(c)

If the distribution functions $F$ and $G$ are unimodal about the same mode $M$ , then $a F+(1-a) G$ is unimodal about $M$ for any $0 < a < 1$ .