A biased die has probability p of showing a 6. The random variable X counts the number of trials up to and including the trial in which the first 6 is obtained. The random variable I counts the number of trials up to and including the trial in which the nth 6 is obtained.

a. Write down an expression for P (X = x) for X = 1. 2, .... Show that the probability generating function of X is:

G_x(t) = pt(1 – qt)^-1

Where 1 = 1 – p. Hence show that the mean and variance of X are. respectively:

E (X) = 1/p

Var(X) = q/p²

b. Given that the trials for X are independent and that Y = X₁ + X₂ + … + X_n, find:

i. The probability generating function for Y

ii. E(Y) and Var(Y).