Expected Values

Prove

differentiate $k$ times with respect to $t$ inside the integral (justify), and derive (21.7) again.

Show that, if $X$ has the standard normal distribution, then $\mathbb {E}\left[|X|^{2 n+1}\right]= 2^{n} n!\sqrt{2 / \pi }$.

$20.9 \uparrow$ Records. Consider the sequence of records in the sense of Problem 20.9. Show that the expected waiting time to the next record is infinite.

20.14 ↑ Show that the Cauchy distribution has no mean.

Prove the first Borel-Cantelli lemma by applying Theorem 16.6 to indicator random variables. Why is Theorem 16.6 not enough for the second Borel-Cantelli lemma?

Prove ( 21.9 ) by Fubini's theorem.

Prove for integrable $X$ that

Suppose that $X$ and $Y$ are random variables with distribution functions $F$ and $G$.

↑ Let $X, Y$, and $Z$ be independent random variables such that $X$ and $Y$ assume the values $0,1,2$ with probability $\frac{1}{3}$ each and $Z$ assumes the values 0 and 1 with probabilities $\frac{1}{3}$ and $\frac{2}{3}$. Let $X^{\prime }=X$ and $Y^{\prime }=X+Z(\bmod 3)$.

Suppose that $X$ and $Y$ are independent, nonnegative random variables and that $\mathbb {E}\left[X\right]=\infty$ and $\mathbb {E}\left[Y\right]=0$. What is the value common to $\mathbb {E}\left[X Y\right]$ and $\mathbb {E}\left[X\right] \mathbb {E}\left[Y\right]$? Use the conventions (15.2) for both the product of the random variables and the product of their expected values. What if $\mathbb {E}\left[X\right]=\infty$ and $0<\mathbb {E}\left[Y\right]<\infty$?

Suppose that $X$ and $Y$ are independent and that $f(x, y)$ is nonnegative. Put $g(x)=\mathbb {E}\left[f(x, Y)\right]$ and show that $\mathbb {E}\left[g(X)\right]=\mathbb {E}\left[f(X, Y)\right]$. Show more generally that $\int_{X \in A} g(X) d P=\int_{X \in A} f(X, Y) d P$. Extend to $f$ that may be negative.

$\uparrow$ The integrability of $X+Y$ does not imply that of $X$ and $Y$ separately. Show that it does if $X$ and $Y$ are independent.

$20.25 \uparrow$ Write $d_{1}(X, Y)=\mathbb {E}\left[\left|X-Y\right| /(1+\left|X-Y\right|)\right]$. Show that this is a metric equivalent to the one in Problem 20.25.

For the density $C \exp \left(-|x|^{1 / 2}\right),-\infty <x<\infty$, show that moments of all orders exist but that the moment generating function exists only at $s=0$.

16.6 ↑ Show that a moment generating function $M(s)$ defined in $\left(-s_{0}, s_{0}\right)$, $s_{0}>0$, can be extended to a function analytic in the strip $\left[z:-s_{0}<\operatorname {Re} z<s_{0}\right]$. If $M(s)$ is defined in $\left[0, s_{0}\right), s_{0}>0$, show that it can be extended to a function continuous in $\left[z: 0 \leq \operatorname {Re} z<s_{0}\right]$ and analytic in $\left[z \cdot 0<\operatorname {Re} z<s_{0}\right]$.

Use (21.28) to find the generating function of (20.39).

For independent random variables having moment generating functions, show by (21.28) that the variances add.

20.17 Show that the gamma density (20.47) has moment generating function $(1-s / \alpha )^{-u}$ for $s<\alpha$. Show that the $k$ th moment is $u(u+1) \cdots (u+k-$ 1) $/ \alpha^{k}$ Show that the chi-squared distribution with $n$ degrees of freedom has mean $n$ and variance $2 n$.

Let $X_{1}, X_{2}, \ldots$ be identically distributed random variables with finite second moment. Show that $n \mathbb {P}\left(\left|X_{1}\right| \geq \epsilon \sqrt{n}\right) \rightarrow 0$ and $n^{-1 / 2} \max_{k \leq n}\left|X_{k}\right| \rightarrow_{p} 0$.