Math Problems Vault

Problem 3.2.1

Let $X$ and $Y$ be independent random variables, each taking the values $-1$ or $1$ with probability $\frac{1}{2}$ , and let $Z = X Y$ . Show that $X, Y$ , and $Z$ are pairwise independent. Are they independent?

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Problem 3.2.2

Let $X$ and $Y$ be independent random variables taking values in the positive integers and having the same mass function $f(x) = 2^{-x}$ for $x = 1,2, \ldots$ . Find:

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(a)

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

(b)

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

(c)

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

(d)

$\mathbb {P}\left(X \geq k Y\right)$ , for a given positive integer $k$ ,

(e)

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

(f)

$\mathbb {P}\left(X = r Y\right)$ , for a given positive rational $r$ .

Problem 3.2.3

(a)(b)

Let $X_{1}, X_{2}, X_{3}$ be independent random variables taking values in the positive integers and having mass functions given by $\mathbb {P}\left(X_{i} = x\right) = \left(1-p_{i}\right) p_{i}^{x-1}$ for $x = 1,2, \ldots$ , and $i = 1,2,3$ .

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(a)

Show that

\mathbb {P}\left(X_{1} < X_{2} < X_{3}\right) = \frac{\left(1-p_{1}\right)\left(1-p_{2}\right) p_{2} p_{3}^{2}}{\left(1-p_{2} p_{3}\right)\left(1-p_{1} p_{2} p_{3}\right)}

(b)

Find $\mathbb {P}\left(X_{1} \leq X_{2} \leq X_{3}\right)$ .

Problem 3.2.4

(a)(b)

Three players, $\mathrm{A}, \mathrm{B}$ , and C, take turns to roll a die; they do this in the order ABCABCA....

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(a)

Show that the probability that, of the three players, $A$ is the first to throw a $6$ , $B$ the second, and C the third, is $216 / 1001$ .

(b)

Show that the probability that the first 6 to appear is thrown by A, the second 6 to appear is thrown by B, and the third 6 to appear is thrown by C, is $46656 / 753571$ .

Problem 3.2.5

Let $X_{r}, 1 \leq r \leq n$ , be independent random variables which are symmetric about 0 ; that is, $X_{r}$ and $-X_{r}$ have the same distributions. Show that, for all $x, \mathbb {P}\left(S_{n} \geq x\right) = \mathbb {P}\left(S_{n} \leq -x\right)$ where $S_{n} = \sum_{r = 1}^{n} X_{r}$ .

Is the conclusion necessarily true without the assumption of independence?

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