8. If X"Bin(n,p) so that P(X =x) = "C.p q" * for x= 0,1,2....,n where q = 1-p , show that: (i) EP(X = x) =1 [Hint: Use binomial theorem expansion] x-0 E(X) = np (iii) EX(X-1) = p*n(n-1) . Therefore find an expression for V(X). (iv) Find M. (t) , the moment generating function. 9. A company wishes to maximize the likelihood _ of winning a specified number of bids for contracts. Suppose that the number of contracts for which bids are made is n = 5 and p represents the chances of winning any given bid for a contract. If X = 3 contracts are to be won, (i) Show that the likelihood is L(P) P ( 1 - p ) 2 . Find the value of p that maximizes the likelihood of this result. Using the general expression in terms of x and n, find an expression for the value of p that maximizes L(p) 10. Twenty percent of men in the population tend to have a particular disease. (i) If 12 men from the population are randomly selected, find the probability that either 2 or 3 of the men have the disease. (ii) A group of men are chosen then tested for the disease. How many men should be chosen if we want a sample of 5 men with the disease