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$P\{X=0\}=3P\{X=3\}$, find $E\left[X\right].$

There are N distinct types of coupons, and each time one is obtained it will, independently of past choices, be of type i with probability Pi, i = 1, ... , N. Let T denote the number one need select to obtain at least one of each type. Compute P{T = n}.

Let$X$ be a binomial random variable with parameters $n$ and $p$. Show that$E\left[\frac{1}{X+1}\right]=\frac{1-(1-p{)}^{n+1}}{(n+1)p}$

Let$X$be the winnings of a gambler. Let $p\left(i\right)=$$P(X=i)$and suppose that

$p\left(0\right)=1/3;p\left(1\right)=p(-1)=13/55$

$p\left(2\right)=p(-2)=1/11;p\left(3\right)=p(-3)=1/165$

Compute the conditional probability that the gambler wins $i,i=1,2,3,$given that he wins a positive amount.

An urn contains$n$balls numbered $1$through $n$. If you withdraw $m$balls randomly in sequence, each time replacing the ball selected previously, find$P\{X=k\},k=$$1,...,m$

where $X$is the maximum of the $m$chosen numbers.

Hint: First find $P\{X≤k\}$.

Teams A and B play a series of games, with the first team to win 3 games being declared the winner of the series. Suppose that team A independently wins each game with probability p. Find the conditional probability that team A wins

(a) the series given that it wins the first game;

(b) the first game given that it wins the series.

In the game of Two-Finger Morra, $2$players show $1$or $2$fingers and simultaneously guess the number of fingers their opponent will show. If only one of the players guesses correctly, he wins an amount (in dollars) equal to the sum of the fingers shown by him and his opponent. If both players guess correctly or if neither guesses correctly, then no money is exchanged. Consider a specified player, and denote by X the amount of money he wins in a single game of Two-Finger Morra.

(a) If each player acts independently of the other, and if each player makes his choice of the number of fingers he will hold up and the number he will guess that his opponent will hold up in such a way that each of the $4$possibilities is equally likely, what are the possible values of $X$and what are their associated probabilities?

(b) Suppose that each player acts independently of the other. If each player decides to hold up the same number of fingers that he guesses his opponent will hold up, and if each player is equally likely to hold up $1$or $2$ fingers, what are the possible values of$X$ and their associated probabilities?

There are n components lined up in a linear arrangement. Suppose that each component independently functions with probability p. What is the probability that no 2 neighboring components are both nonfunctional?

A local soccer team has $5$more games left to play. If it wins its game this weekend, then it will play its final $4$games in the upper bracket of its league, and if it loses, then it will play its final games in the lower bracket. If it plays in the upper bracket, then it will independently win each of its games in this bracket with probability .$4$, and if it plays in the lower bracket, then it will independently win each of its games with probability .$7$. If the probability that the team wins its game this weekend is .$5$, what is the probability that it wins at least$3$of its final$4$games?

Let X be a binomial random variable with parameters (n, p). What value of p maximizes P{X = k}, k = 0, 1, ... , n? This is an example of a statistical method used to estimate p when a binomial (n, p) random variable is observed to equal k. If we assume that n is known, then we estimate p by choosing that value of p that maximizes P{X = k}. This is known as the method of maximum likelihood estimation.