91Ó°ÊÓ

The random variable \(X\) has the following probability mass function $$ p(1)=\frac{1}{2}, \quad p(2)=\frac{1}{3}, \quad p(24)=\frac{1}{6} $$ Calculate \(E[X]\)

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 \(p=1 / 2\).

An urn contains \(n+m\) balls, of which \(n\) are red and \(m\) are black. They are withdrawn from the um, one at a time and without replacement. Let \(X\) be the number of red balls removed before the first black ball is chosen. We are interested in determining \(E[X]\). To obtain this quantity, number the red balls from 1 to \(n\). Now define the random variables \(X_{i}, i=1, \ldots, n\), by $$ X_{i}=\left\\{\begin{array}{ll} 1, & \text { if red ball } i \text { is taken before any black ball is chosen } \\ 0, & \text { otherwise } \end{array}\right. $$ (a) Express \(X\) in terms of the \(X_{i}\). (b) Find \(E[X]\).

A total of \(r\) keys are to be put, one at a time, in \(k\) boxes, with each key independently being put in box \(i\) with probability \(p_{i}, \sum_{i=1}^{k} p_{i}=1 .\) Each time a key is put in a nonempty box, we say that a collision occurs. Find the expected number of collisions.

Consider three trials, each of which is either a success or not. Let \(X\) denote the number of successes. Suppose that \(E[X]=1.8\). (a) What is the 'largest possible value of \(P\\{X=3\\}\) ? (b) What is the smallest possible value of \(P[X=3\\} ?\) In both cases, construct a probability scenario that results in \(P\\{X=3\) \\} having the desired value.

Let \(c\) be a constant. Show that (i) \(\operatorname{Var}(c X)=c^{2} \operatorname{Var}(X)\). (ii) \(\operatorname{Var}(c+X)=\operatorname{Var}(X)\) \(\because\)

There are \(n\) types of coupons. Each newly obtained coupon is, independently, type \(i\) with probability \(p_{i}, i=1, \ldots, n\). Find the expected number and the variance of the number of distinct types obtained in a collection of \(k\) coupons.

Suppose that \(X\) and \(Y\) are independent binomial random variables with parameters \((n, p)\) and \((m, p)\). Argue probabilistically (no computations necessary) that \(X+Y\) is binomial with parameters \((n+m, p)\).

Calculate the moment generating function of the uniform distribution on \((0,1) .\) Obtain \(E[X]\) and \(\operatorname{Var}[X]\) by differentiating.

Show that the sum of independent identically distributed exponential random variables has a gamma distribution.