# Question: Consider n independent trials each resulting in any one of

Consider n independent trials, each resulting in any one of r possible outcomes with probabilities P1, P2, . . . , Pr. Let X denote the number of outcomes that never occur in any of the trials. Find E[X] and show that, among all probability vectors P1, . . . ,Pr, E[X] is minimized when Pi = 1/r, i = 1, . . . , r.

## Answer to relevant Questions

Suppose that each of the elements of S = {1, 2, . . . , n} is to be colored either red or blue. Show that if A1, . . . ,Ar are subsets of S, there is a way of doing the coloring so that at most of these subsets have all ...The Conditional Covariance Formula. The conditional covariance of X and Y, given Z, is defined by Cov(X, Y|Z) = E[(X − E[X|Z])(Y − E[Y|Z])|Z] (a) Show that Cov(X, Y|Z) = E[XY|Z] − E[X|Z]E[Y|Z] (b) Prove the conditional ...Consider Example 4f, which is concerned with the multinomial distribution. Use conditional expectation to compute E[NiNj], and then use this to verify the formula for Cov(Ni,Nj) given in Example 4f. Civil engineers believe that W, the amount of weight (in units of 1000 pounds) that a certain span of a bridge can withstand without structural damage resulting, is normally distributed with mean 400 and standard deviation ...Use the central limit theorem to solve part (c) of Problem 2. Part (c) of Problem 2 How many students would have to take the examination to ensure, with probability at least .9, that the class average would be within 5 of ...Post your question