Chapter 5.9, Problem 119E

A learning experiment requires a rat to run a maze (a network of pathways) until it locates one of three possible exits. Exit 1 presents a reward of food, but exits 2 and 3 do not. (If the rat eventually selects exit 1 almost every time, learning may have taken place.) Let Y_i denote the number of times exit i is chosen in successive runnings. For the following, assume that the rat chooses an exit at random on each run.

a Find the probability that n = 6 runs result in Y₁ = 3, Y₂ = 1, and Y₃ = 2.

b For general n, find E(Y₁) and V(Y₁).

c Find Cov(Y₂, Y₃) for general n.

d To check for the rat’s preference between exits 2 and 3, we may look at Y₂ − Y₃. Find E(Y₂ − Y₃) and V(Y₂ − Y₃) for general n.