Question

Suppose you flip a fair coin six times where A is the number of tails and B is the number of heads. Prove if E(X*Y) is not equal to E(X)E(Y).

Answer 1

The joint probabilty distribution of A and B is given as,

P(A = a, B = b) ~ Binomial(n = 6, p = 0.5) for a + b = 6

P(A = a, B = b) = 0 for a + b 6

= 5 * 0.09375 + 8 * 0.234375 + 9 * 0.3125 + 8 * 0.234375 + 5 * 0.09375

= 7.5

Also, A ~ Binomial(n = 6, p = 0.5) and B ~ Binomial(n = 6, p = 0.5)

E(A) = np = 6 * 0.5 = 3

E(B) = np = 6 * 0.5 = 3

E(A) * E(B) = 9

Thus, E(AB) E(A) E(B)

Thus, E(X*Y) is not equal to E(X)E(Y) for A and B.