Chapter 5: Special Random Variables 17. IfX is a Poisson random variable with mean 2, show that P{X=i} first increases and then decreases as i increases, reaching its maximum value when i is the largest 000 18. A contractor purchases a shipment of 100 transistors. It is his policy to test 10 of these transistors and to keep the shipment only if at least 9 of the working condition. If the shipment contains 20 defective transistors, what is the integer less than or equal to 2. probability it will be kept? 19. Let X denote a hypergeometric random variable with parameters n, m, and k. That is, ()C) (w) i = 0, 1,..., min(k, n) P{X = i} = %3D (a) Derive a formula for P{X = i} in terms of P{X =i – 1}. (b) Use part (a) to compute P{X = i} for i = 0, 1, 2, 3, 4, 5 when n = m = 10. k = 5, by starting with P{X = 0}. (c) Based on the recursion in part (a), write a program to compute the hyper- %3D %3D 10, %3D %3D %3D geometric distribution function. (d) Use your program from part (c) to compute P{X < 10} when n = m= 30, k = 15. 20. Independent trials, each of which is a success with probability p, are successively performed. Let X denote the first trial resulting in a success. That is, X will equal k if the first k-1 trials are all failures and the kth a success. X is called a geometric random variable. Compute (a) P{X= k}, k = 1, 2, ...; (b) E[X]. %3D Let Y denote the number of trials needed to obtain r successes. Y is called a negative binomial random variable. Compute (c) P(Y = k}, k= r,r+1,.... (Hint: In order for Y to equal k, how many successes must result in the first k: trials and what must be the outcome of trial k?) (d) Show that E[Y] = rlp (Hint: Write Y = Y1++ Y, where Y; is the number of trials needed to go from a total of i-1 to a total of i successes.)