The multinomial distribution. A die with r faces, numbered

QUESTION:

The multinomial distribution. A die with r faces, numbered 1, .... r. is rolled a fixed number of times n. The probability that the ith face comes up on any one roll is denoted pz , and the results of different rolls are assumed independent. Let Xz be the number of times that the ith face comes up. (a) Find the joint PMF pxl ..... xr (k1, . , kr). (b) Find the expected value and variance of Xi . (c) Find E[XlXJJ for i - j. Solution. (a) The probability of a sequence of rolls where, for i = 1, . . . , r, face i comes up ki times is pl ... pr . Every such sequence determines a partition of the set of n rolls into r subsets with the ith subset having cardinality ki (this is the set of rolls for which the ith face came up). The number of such partitions is the multinomial coefficient (cf. Section 1.6) ( n ) n! kl ' . . . , kr kl ! . . . kr! Thus, if kl + ... + kr = n, and otherwise, Pxl . . . . . xr(kl , . . . , kr) = O. (b) The random variable Xl is binomial with parameters n and Pt. Therefore, E[Xi] = npi, and var(Xd = npi (l - pd. (c) Suppose that i =f::. j, and let Yik (or Yj,k ) be the Bernoulli random variable that takes the value 1 if face i (respectively, j) comes up on the kth roll. and the value 0 otherwise. Note that Yi,kYj,k = 0, and that for l =f::. k. Yt.k and Yj.1 are independent, so that E[Yi .k Yj,l] = PiPj. Therefore, E[XlXj] = E [(Yi. l + . . . + Yt.n)(Y). 1 + ... + Yj.n)] = n(n l)E[Yi, 1 Yj,2] = n(n - l)PiPj.