1. 4.12 Consider an experiment with three possible outcomes 1,2, and 3, which occur with probabilities of p1, p2, and p3 = 1 − p1 − p2. For a given value of n, n independent trials of the experiment are performed. The random variable Xi gives the number of times that the outcome i occurs for i = 1,2,3. Verify that P(X1 = k1, X2 = k2) = n k1 n−k1 k2 pk1 1 pk2 2 pn−k1−k2 3

for all k1,k2 ≥ 0 with k1 +k2 ≤ n. This distribution is called the multinomial distribution (with r = 3 possible outcomes).