Suppose we consider a trial that can have l possible outcomes (say, rolling an l-sided die). Each of the outcomes occurs with some probability P, P2, ..., pe with p+p2+...+pe = 1. Suppose we have n independent trials. Let Z (Z,... ..., Ze) be the random vector specifying the number of times each outcome occurs. (a) Show that we have n ki P P(Z = (k,..., kt)) = (k, ..., k.). p^ ke 2 ke Pe = n! k k!... k... pke, where k+...+ke = n. This is known as the the Multinomial distribution. (b) Show that the marginal distribution of each Z; is given by a binomial distribution with parameter p. (Hint: you can do this without computing any nasty sums.)