Biased Coin Toss
A biased coin is tossed repeatedly, with the probability of getting a head equal to p (0<p<1) and the result of each toss being independent of any other toss. Let X1 be the number of heads before the first tail and X2 be the number of heads between the first and second tail. By showing that
P(X1=j, X2=k) = P(X1=j) x P(X2=k)
(i) Verify that X1 and X2 are independent.
(ii) Derive the p.g.f. of Xi (i=1 or 2)
(iii) find the p.g.f of X1+X2 and hence determine the probability distribution of X1+X2
Thank you, any help appreciated