A box contains b blue and r red balls (total number of balls in the box is n = b + r). All balls are removed at random one by one and arranged in a row. Let X, be the number of red balls between the (i - 1)th and ith blue ball drawn, i = 2, ..., b; Let X ] be the number of red balls until the first blue ball shows up, and Xb+ 1 be the number of red balls after the last blue ball drawn. In #5 of hw4, we found the pmf of X" : P(X = (k1 . . . ..kb+1)) = P(X1 = ki , . . .. Xb+1 = kb+1) (6.3) bir! = f ( KI... . .kb+1) = n! for any nonnegative integers k1, . .. . kb+ 1 so that k1 + .. . + kb+1= r. a) Find E(X1). .... E(Xb+1) . Hint. The pmf f of X in (6.3) is symmetric in (k1, . .. .kb+1): if we shuffle (rearrange) 1, ....kb+1 the value of f does not change (it is still f (k1, . ... kb+ 1)). Therefore all the components X1, ..., Xy+ 1 are identically distributed (they have the same pmf, that is all marginal pmfs are identical). Why? b) Let Y, be the number of balls needed to be removed until the ith blue ball shows up, i = 1, . .., b. Find E (Y,) , i = 1, ..., b. Hint. Y1 = X1+ 1 c) Find the pmf of X] and Y1. What are the pmf of X2. ..., Xb+