Consider independent trials, each of which results in outcome 0, 1 or 2, with the respective probabilities 1/2, 1/4 and 1/4. Let N denote the number of trials needed to obtain an outcome that is not equal to 2, and let X be that outcome.

(a) FindP(N=n),n1

(b) FindP(X=j),j=0,1

(c) Show that N and X are independent, that is, P(N = n,X = j) = P(N = n)P(X = j).