Math Problems Vault

Problem 2.1.1

(a)(b)

Let $X$ be a random variable on a given probability space, and let $a \in \mathbb {R}$ . Show that

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(a)

$a X$ is a random variable,

(b)

$X-X = 0$ , the random variable taking the value 0 always, and $X+X = 2 X$ .

Problem 2.1.2

A random variable $X$ has distribution function $F$ . What is the distribution function of $Y = a X+b$ , where $a$ and $b$ are real constants?

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Problem 2.1.3

A fair coin is tossed $n$ times. Show that, under reasonable assumptions, the probability of exactly $k$ heads is $\binom {n}{k}\left(\frac{1}{2}\right)^{n}$ . What is the corresponding quantity when heads appears with probability $p$ on each toss?

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Problem 2.1.4

Show that if $F$ and $G$ are distribution functions and $0 \leq \lambda \leq 1$ then $\lambda F+(1-\lambda ) G$ is a distribution function. Is the product $F G$ a distribution function?

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Problem 2.1.5

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

Let $F$ be a distribution function and $r$ a positive integer. Show that the following are distribution functions:

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(a)

$F(x)^{r}$ ,

(b)

$1-\left\{ 1-F(x)\right\}^{r}$ ,

(c)

$F(x)+\left\{ 1-F(x)\right\} \log \left\{ 1-F(x)\right\}$ ,

(d)

$\left\{ F(x)-1\right\} e+\exp \left\{ 1-F(x)\right\}$ .

Problem 2.1.6

A random variable that is equally likely to take any value in a finite set $S$ is said to have the uniform distribution on $S$ . If $U$ is such a random variable and $\varnothing \neq R \subseteq S$ , show that the distribution of $U$ conditional on $\left\{ U \in R\right\}$ is uniform on $R$ .

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