91Ó°ÊÓ

Suppose that the random variable \(X\) is equal to the number of hits obtained by a certain, baseball player in his next 3 at bats. If \(P\\{X=1\\}=.3\), \(P\\{X=2\\}=.2\), and \(P\\{X=0\\}=3 P\\{X=3\\}\), find \(E[X]\).

Two fair dice are rolled. Let \(X\) equal the product of the 2 dice. Compute \(P\\{X=i\\}\) for \(i=1,2, \ldots\)

A coin that when flipped comes up heads with probability \(p\) is flipped until either heads or tails has occurred twice. Find the expected. number of flips.

A certain community is composed of \(m\) families, \(n_{i}\) of which have \(i\) children, \(\sum_{i=1}^{r} n_{i}=m .\) If one of the families is randomly chosen, let \(X\) denote the number of children in that family. If one of the \(\sum_{i=1}^{r}\) in \(_{l}\) children is randomly chosen, let \(Y\) denote the total number of children in the family of that child. Show that \(E[Y] \geq E[X]\).

If \(X\) has distribution function \(F\), what is the distribution function of the random variable \(\alpha X+\beta\), where \(\alpha\) and \(\beta\) are constants, \(\alpha \neq 0\) ?

For a nonnegative integer-valued random variable \(N\), show that $$ E[N]=\sum_{i=1}^{\infty} P\\{N \geq i\\} $$

Consider \(n\) independent sequential trials, each of which is successful with probability \(p\). If there is a total of \(k\) successes, show that each of the \(n ! /[k !(n-k) !]\) possible arrangements of the \(k\) successes and \(n-k\) failures is equally likely.

Five distinct numbers are randomly distributed to players numbered 1 through 5\. Whenever two players compare their numbers, the one with the higher one is declared the winner. Initially, players 1 and 2 compare their numbers; the winner then compares with player 3, and so on. Let \(X\) denote the number of times player 1 is a winner. Find \(P\\{X=i\\}, i=0,1,2,3,4\).

Four independent flips of a fair coin are made. Let \(X\) denote the number of heads obtained. Plot the probability mass function of the random variable \(X-2\)

A total of 4 buses carrying 148 students from the same school arrives at a football stadium. The buses carry, respectively, \(40,33,25\), and 50 students. One of the students is randomly selected. Let \(X\) denote the number of students that were on the bus carrying this randomly selected student. One of the 4 bus drivers is also randomly selected. Let \(Y\) denote the number of students on her bus. (a) Which of \(E[X]\) or \(E[Y]\) do you think is larger? Why? (b) Comnute \(F[X]\) and \(E[Y)\)