91Ó°ÊÓ

A pond contains 100 fish, of which 30 are carp. If 20 fish are caught, what are the mean and variance of the number of carp among the \(20 ?\) What assumptions are you making?

A group of 20 people consisting of 10 men and 10 women is randomly arranged into 10 pairs of 2 each. Compute the expectation and variance of the number of pairs that consist of a man and a woman. Now suppose the 20 people consist of 10 married couples. Compute the mean and variance of the number of married couples that are paired together.

A fair die is successively rolled. Let \(X\) and \(Y\) denote, respectively, the number of rolls necessary to obtain a 6 and a \(5 .\) Find (a) \(E[X]\) (b) \(E[X | Y=1]\) (c) \(E[X | Y=5]\)

A coin having probability \(p\) of coming up heads is continually flipped until both heads and tails have appeared. Find (a) the expected number of flips; (b) the probability that the last flip lands on heads.

Let \(X_{1}, \ldots\) be independent random variables with the common distribution function \(F,\) and suppose they are independent of \(N,\) a geometric random variable with parameter \(p .\) Let \(M=\max \left(X_{1}, \ldots, X_{N}\right)\) (a) Find \(P\\{M \leq x\\}\) by conditioning on \(N\) (b) Find \(P\\{M \leq x | N=1\\}\) (c) Find \(P\\{M \leq x | N>1\\}\) (d) Use (b) and (c) to rederive the probability you found in (a).