Question 1 (a) The Binomial distribution has probability mass function given by: P(X = x) = "Cp*q"-* x=0, 1, 2,.., n (i) Derive the moment generating function (ii) Show that the expected value is np (b) A game is played such that trials are made repeatedly until an attempt results in the first triumph. The probability that this player scores a triumph on any given trial is 0.2. If the game is played only 4 times (i) Find the probability that the triumph is scored after the second trial in exactly one of the 4 games played. (ii) Find the expected number of games in which the triumph is made after the second trial. Hint: This question combines Binomial probability distribution and Geometric probability distribution