Consider an unfair coin that turns up heads one-fourth of the time. I toss the coin three times

Question

Consider an unfair coin that turns up heads one-fourth of the time. I toss the coin three times

and let X equal the number of heads. Find the probability distribution of X and then find its mean and variance. [ Hint: If X ∼Bin (n, p) then E(X) =np and Var(X) =np(1−p)]

Answer 1

x probability P(x)

0 TTT 3/4 * 3/4 * 3/4 = 27/64 27/64

1 HTT 1/4*3/4*3/4 = 9/64 27/64

THT 9/64

TTH 9/64

2 HHT 1/4 * 1/4 * 3/4 = 3/64 9/64

THH 3/64

HTH 3/64

3 HHH 1/4 * 1/4 */14 1/64

E(x) = 3 * 1/4 = 3/4

V(x) = 3 * 1/4 * 3/4 = 9/16