25. A variation of the binomial distribution arises when

the n trials are all independent, but the probability of a success on the ith trial is i, and these probabilities are

not all equal. If X is the number of successes obtained

under these conditions in n trials, show that

(a) X = n, where = 1n ni=1 i;

(b) 2X = n(1 ) n2 , where is as defined in part

(a) and 2 = 1n ni=1

(i )2.