Probability sums

Page 164, 3.1.12." Let X1. X2. . . ., Xk-1 have a multinomial distribution. (a) Find the mgf of X1, X3. . . . . Xk-1. (b) What is the pmf of X1, X3. . . .,Xk-1? (c) Determine the conditional pmf of X2 given that X1= X1, X3 = X3, . . . . Xk-1 = Xk-1. (d) What is the conditional expectation E(X2|x1,*3. . . . . *k-1)? Page 167, 3.1.30.* Consider a shipment of 1000 items into a factory. Suppose the factory can tolerate about 4% defective items. Let X be the number of defective items in a sample without replacement of size n = 10. Suppose the factory returns the shipment if X 2 1. (a) Obtain the probability that the factory DOES NOT return a shipment of items that has 4% defective items. (b) Suppose the shipment has 8% defective items. Obtain the probability that the factory returns such a shipment. (c) Obtain approximations to the probabilities in parts (a) and (b) using appropriate binomial distributions. Page 173, 3.2.16.* Let X and Y and have joint probability mass function (p. m. f.) given by f (x, y) = {x!(v-x)! = 0, 1, .. ., co, x = 0,1,2, . .., y. 0. elsewhere. (a) Find the joint moment generating function M(t1, t2). (b) Compute the correlation coefficient of X and Y. Page 184, 3.3.14." In a warehouse of parts for a large mill, the average time between requests for parts is about 8 minutes. (a) Find the probability that in an hour there will be at most 5 requests for parts. (b) Find the probability that the 12th request in the morning requires at least 2 hours of waiting time. Page 197, 3.4.30.* Compute P([X, + 2X2 - 3X3]? > 49) if X1, X2, Xy are i. i. d. with common distribution N(1, 1)