Chapter 2: Problem 34
Let the probability density of \(X\) be given by
$$
f(x)=\left\\{\begin{array}{ll}
c\left(4 x-2 x^{2}\right), & 0
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If \(X\) is a nonnegative random variable, and \(g\) is a differential function with \(g(0)=0\), then $$ E[g(X)]=\int_{0}^{\infty} P(X>t) g^{\prime}(t) d t $$ Prove the preceding when \(X\) is a continuous random variable.
Suppose that two teams are playing a series of games, each of which is independently won by team \(A\) with probability \(p\) and by team \(B\) with probability \(1-p .\) The winner of the series is the first team to win four games. Find the expected number of games that are played, and evaluate this quantity when \(p=1 / 2\).
If \(X\) is a nonnegative integer valued random variable, show that (a) $$ E[X]=\sum_{n=1}^{\infty} P[X \geq n\\}=\sum_{n=0}^{\infty} P(X>n\\} $$ Hint: Define the sequence of random variables \(I_{n}, n \geq 1\), by $$ I_{n}=\left\\{\begin{array}{ll} 1, & \text { if } n \leq X \\ 0, & \text { if } n>X \end{array}\right. $$ Now express \(X\) in terms of the \(I_{n}\). (b) If \(X\) and \(Y\) are both nonnegative integer valued random variables, show that $$ E[X Y]=\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} P(X \geq n, Y \geq m) $$
Let \(X\) denote the number of white balls selected when \(k\) balls are chosen at random from an urn containing \(n\) white and \(m\) black balls. (a) Compute \(P[X=i\\}\). (b) Let, for \(i=1,2, \ldots, k ; j=1,2, \ldots, n\) \(X_{i}=\left\\{\begin{array}{ll}1, & \text { if the } i \text { th ball selected is white } \\ 0, & \text { otherwise }\end{array}\right.\) \(Y_{j}=\left\\{\begin{array}{ll}1, & \text { if white ball } j \text { is selected } \\ 0, & \text { otherwise }\end{array}\right.\) Compute \(E[X]\) in two ways by expressing \(X\) first as a function of the \(X_{i} s\) and then of the \(Y_{j}\) s.
A television store owner figures that 50 percent of the customers entering his store will purchase an ordinary television set, 20 percent will purchase a color television set, and 30 percent will just be browsing. If five customers enter his store on a certain day, what is the probability that two customers purchase color sets, one customer purchases an ordinary set, and two customers purchase nothing?
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