91Ó°ÊÓ

Prove the Cauchy-Schwarz inequality, namely, that $$ (E[X Y])^{2} \leq E\left[X^{2}\right] E\left[Y^{2}\right] $$ HINT: Unless \(Y=-t X\) for some constant, in which case this inequality holds with equality, if follows that for all \(t\), $$ 0

If \(X_{1}, X_{2}, \ldots, X_{n}\) are independent and identically distributed random variables having uniform distributions over \((0,1)\), find (a) \(E\left[\max \left(X_{1}, \ldots, X_{n}\right)\right] ;\) (b) \(E\left[\min \left(X_{1}, \ldots, X_{n}\right)\right]\)

Prove that \(E[g(X) Y \mid X]=g(X) E[Y \mid X]\).

Prove that if \(E[Y \mid X=x]=E[Y]\) for all \(x\), then \(X\) and \(Y\) are uncorrelated, and give a counterexample to show that the converse is not true. HINT: Prove and use the fact that \(E[X Y]=E[X E[Y \mid X]]\).

An urn contains \(a\) white and \(b\) black balls. After a ball is drawn, it is retumed to the urn if it is white; but if it is black, it is replaced by a white ball from another urn. Let \(M_{n}\) denote the expected number of white balls in the um after the foregoing operation has been repeated \(n\) times. (a) Derive the recursive equation $$ M_{n+1}=\left(1-\frac{1}{a+b}\right) M_{n}+1 $$ (b) Use part (a) to prove that $$ M_{n}=a+b-b\left(1-\frac{1}{a+b}\right)^{n} $$ (c) What is the probability that the \((n+1)\) st ball drawn is white?

If \(X_{1}, X_{2}, X_{3}, X_{4}\) are (pairwise) uncortelated random variables each having mean 0 and variance 1 , compute the correlations of (a) \(X_{1}+X_{2}\) and \(X_{2}+X_{3}\); (b) \(X_{1}+X_{2}\) and \(X_{3}+X_{4}\)

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) \(E[X \mid Y=1]\); (c) \(E[X \mid Y=5]\).

An urn contains 4 white and 6 black balls. Two successive random samples of sizes 3 and 5 , respectively, are drawn from the urn without replacement. Let \(X\) and \(Y\) denote the number of white balls in the two samples, and compute, \(E[X \mid Y=i]\), for \(i=1,2,3,4\)

Consider the following dice game. A pair of dice are rolled. If the sum is 7 , then the game ends and you win 0 . If the sum is not 7 , then you have the option of either stopping the game and receiving an amount equal to that sum or starting over again. For each value of \(i, i=2, \ldots, 12\), find your expected return if you employ the strategy of stopping the first time that a value at least as large as \(i\) appears. What value of \(i\) leads to the largest expected return? HINT: Let \(X_{i}\) denote the return when you use the critical value \(i\). To compute \(E\left[X_{i}\right]\), condition on the initial sum:

The number of people that enter an elevator on the ground floor is a Poisson random variable with mean 10. If there are \(N\) floors above the ground floor and if each person is equally likely to get off at any one of these \(N\) floors, independently of where the others get off, compute the expected number of stops that the elevator will make before discharging all of its passengers.