Answer these questions. Kindly include step by step explanations.

In a process, the number of failures before the first success has the following probability distribution: P(X = x) = 0(1-0)* x = 0,1,2.... Show that the method of moments and the method of maximum likelihood both lead to the same estimator for 0 . [5]The random variable Y has a gamma distribution with parameters o (>1) and 1. (i) (a) Show that the mode of Y is given by: a -1 (b) By considering the relative locations of the mean and mode using sketches of the gamma distribution, state how you would expect the distribution to behave in the limit as or -> co, but where 1 is varied so that the mean has a constant value (. [5] (ii) Given that o = 50 and 1 = 0.2, calculate the value of P(Y > 350) using: (a) the gamma-chi square relationship (b) the Central Limit Theorem. [5] (iii) Explain the reason for the difference between the answers obtained in part (ii) [1] [Total 1 1]The random variable Y is defined by: Z Y =>X, i=1 where the X,'s are independent Bin(n, p) random variables and N is a Poisson random variable, independent of the X, 's, with mean 1. (i) Prove from first principles that the moment generating function of Y is given by: My(1) = expo 2 (q + pe')" -1]} [4] (ii) Hence derive expressions for the mean and variance of Y . [4] [Total 8]