Chapter 1: Problem 20
Let \(U\) be gamma distributed with order \(p\) and let \(V\) have the beta
distribution with parameters \(q\) and \(p-q(0 Over 30 million students worldwide already upgrade their
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Suppose we have \(N\) chips, numbered \(1,2, \ldots, N .\) We take a random sample of size \(n\) without replacement. Let \(X\) be the largest number in the random sample. Show that the probability function of \(X\) is $$ \operatorname{Pr}\\{X=k\\}=\frac{\left(\begin{array}{l} k-1 \\ n-1 \end{array}\right)}{\left(\begin{array}{c} N \\ n \end{array}\right)} \quad \text { for } k=n, n+1, \ldots, N $$ and that $$ E X=\frac{n}{n+1}(N+1), \quad \operatorname{Var}(X)=\frac{n(N-n)(N+1)}{(n+1)^{2}(n+2)} $$
Consider an infinite number of urns into which we toss balls independently, in such a way that a ball falls into the \(k\) th urn with probability \(1 / 2^{k}, k=1,2,3\). \(\ldots .\) For each positive integer \(N\), let \(Z_{N}\) be the number of urns which contain at least one ball after a total of \(N\) balls have been tossed. Show that $$ E\left(Z_{N}\right)=\sum_{\lambda=1}^{\infty}\left[1-\left(1-1 / 2^{k}\right)^{N}\right] $$ and that there exist constants \(C_{1}>0\) and \(C_{2}>0\) such that $$ C_{1} \log N \leq E\left(Z_{N}\right) \leq C_{2} \log N \quad \text { for all } N $$ Hint: Verify and use the facts: $$ E\left(Z_{N}\right) \geq \sum_{k=1}^{\log _{2}-N}\left[1-\left(1-\frac{1}{2^{k}}\right)^{N}\right] \geq C \log _{2} N $$ and $$ 1-\left(1-\frac{1}{2^{k}}\right)^{N} \leq N \frac{1}{2^{k}} \text { and } N \sum_{\log _{2}}^{\infty} \frac{1}{2^{k}} \leq C_{2} $$
Let \(X\) be a nonnegative random variable with cumulative distribution funetion \(F(x)=\operatorname{Pr}\\{X \leq x\\}\). Show $$ E[X]=\int_{0}^{\infty}[1-F(x)] d x $$ Hint: Write \(E[X]=\int^{\infty} x d F(x)=\int^{\infty}\left(\int_{0}^{x} d y\right) d F(x)\).
The random variables \(X\) and \(Y\) have the following properties: \(X\) is positive, i.e., \(P\\{X>0\\}=1\), with continuous density funetion \(f(x)\), and \(Y \mid X\) has a uniform distribution on \(\\{0, X\\} .\) Prove: If \(Y\) and \(X-Y\) are independently dis* tributed, then $$ f(x)=a^{2} x e^{-a x}, \quad x>0, \quad a>0 $$
Suppose that a lot consists of \(m, n_{1}, \ldots, n_{r}\), items belonging to the \(0 \mathrm{th}\), (1n1,..., \(r\) th classes respeetively. The items are drawn one-by-one without replace. ment until \(k\) items of the 0 th elass are observed. Show that the joint aistribution of the observed frequencies \(X_{1}, \ldots, X_{r}\) of the Ist,..., \(r\) th classes is $$ \begin{gathered} \operatorname{Pr}\left\\{X_{1}=x_{1}, \ldots, X_{r}=x_{r}\right\\}=\left\\{\left(\begin{array}{c} m \\ k-1 \end{array}\right) \prod_{i=1}^{r}\left(\begin{array}{l} n_{i} \\ x_{i} \end{array}\right) /\left(\begin{array}{c} m+n \\ k+y-1 \end{array}\right)\right\\} \\ \cdot \frac{m-(k-1)}{m+n-(k+y-1)} \end{gathered} $$ where $$ y=\sum_{i=1}^{r} x_{1} \quad \text { and } \quad n=\sum_{i=1}^{r} n_{i^{*}} $$
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