Chapter 3

For continuous random variables \(X\) and \(Y\), prove that \(E(Y)=E_{X}[E(Y \mid x)]\).

Two chips are drawn at random and without replacement from an urn that contains five chips, numbered 1 through 5 . If the sum of the chips drawn is even, the random variable \(X\) equals 5 ; if the sum of the chips drawn is odd, \(X=-3\). Find the moment-generating function for \(X\).

Find the expected value of \(e^{3 X}\) if \(X\) is a binominal random variable with \(n=10\) and \(p=\frac{1}{3}\).

Find the moment-generating function for the discrete random variable \(X\) whose probability function is given by $$ p_{X}(k)=\left(\frac{3}{4}\right)^{k}\left(\frac{1}{4}\right), \quad k=0,1,2, \ldots $$

. Let \(Y\) have pdf $$ f_{Y}(y)= \begin{cases}y, & 0 \leq y \leq 1 \\ 2-y, & 1 \leq y \leq 2 \\ 0, & \text { elsewhere }\end{cases} $$ Find \(M_{Y}(t)\).

The random variable \(X\) has a Poisson distribution \(p_{X}(k)=e^{-\lambda} \lambda^{k} / k !, k=0,1,2, \ldots\). Find the momentgenerating function for a Poisson random variable. Recall that $$ e^{r}=\sum_{k=0}^{\infty} \frac{r^{k}}{k !} $$

Let \(Y\) be a continuous random variable with \(f_{Y}(y)=y e^{-y}, 0 \leq y .\) Show that \(M_{Y}(t)=\frac{1}{(1-t)^{2}} .\)

Calculate \(E\left(Y^{3}\right)\) for a random variable whose moment-generating function is \(M_{Y}(t)=e^{t^{2} / 2}\).

Find \(E\left(Y^{4}\right)\) if \(Y\) is an exponential random variable with \(f_{Y}(y)=\lambda e^{-\lambda y}, y>0\).

The form of the moment-generating function for a normal random variable is \(M_{Y}(t)=e^{a t t+b^{2} t^{2} / 2}\) (recall Example 3.12.4). Differentiate \(M_{Y}(t)\) to verify that \(a=E(Y)\) and \(b^{2}=\operatorname{Var}(Y)\)

What is \(E\left(Y^{4}\right)\) if the random variable \(Y\) has moment-generating function \(M_{Y}(t)=(1-\alpha t)^{-k}\) ?

Find an expression for \(E\left(Y^{k}\right)\) if \(M_{Y}(t)=(1-\) \(t / \lambda)^{-r}\), where \(\lambda\) is any positive real number and \(r\) is a positive integer.

Find the variance of \(Y\) if \(M_{Y}(t)=e^{2 t} /\left(1-t^{2}\right)\).

Calculate \(P(X \leq 2)\) if \(M_{X}(t)=\left(\frac{1}{4}+\frac{3}{4} e^{t}\right)^{5}\)

. Suppose that \(X\) is a Poisson random variable, where \(p_{X}(k)=e^{-\lambda} \lambda^{k} / k !, k=0,1, \ldots\) (a) Does the random variable \(W=3 X\) have a Poisson distribution? (b) Does the random variable \(W=3 X+1\) have a Poisson distribution?

Suppose that \(Y\) is a normal variable, where \(f_{Y}(y)=(1 / \sqrt{2 \pi} \sigma) \exp \left[-\frac{1}{2}\left(\frac{y-\mu}{\sigma}\right)^{2}\right],-\infty