tailieunhanh - The Binomial Probability Distribution

To help you major in English with additional material in the process of learning and working, you are invited to consult the book "The Binomial Probability Distribution" below. Hopefully, the books contents useful service for you. | The Binomial Probability Distribution 135 The Moment Generating Function of X Let s find the moment generating function of a binomial random variable. Using the definition MX t E etX Mx t E etX X txp x etx U 1 - p -x x 2 D x 0 V x X Pet X 1 p n Pet 1 _ p n X 0 Vx Here we have used the binomial theorem 2 0 axb x a b n. Notice that the mgf satisfies the property required of all moment generating functions MX 0 1 because the sum of the probabilities is 1. The mean and variance can be obtained by differentiating MX t MX t n pe 1 p -1pei and m MX 0 np Then the second derivative is M X t 1 pe 1 p 2pe pe pet 1 p -1pef and E X2 MX 0 - 1 p2 np Therefore s2 V X E X2 - E X 2 1 p2 np 2p2 np np2 np 1 p in accord with the foregoing proposition. Exercises Section 58-79 58. Compute the following binomial probabilities directly from the formula for b x n p a. b 3 8 .6 b. b 5 8 .6 c. P 3 X 5 when n 8 and p .6 d. P 1 X when n 12 and p .1 a. b. c. d. e. 59. Use Appendix Table to obtain the following probabilities B 4 10 .3 b 4 10 .3 b 6 10 .7 P 2 X 4 when X Bin 10 .3 P 2 X when X Bin 10 .3 f. P X 1 when X Bin 10 .7 g. P 2 X 6 when X Bin 10 .3 60. When circuit boards used in the manufacture of compact disc players are tested the long-run percentage of defectives is 5 . Let X the number of defective boards in a random sample of size n 25 so X Bin 25 .05 . a. Determine P X 2 . b. Determine P X 5 . c. Determine P 1 X 4 . d. What is the probability that none of the 25 boards is defective 136 CHAPTER 3 Discrete Random Variables and Probability Distributions e. Calculate the expected value and standard deviation of X. 61. A company that produces fine crystal knows from experience that 10 of its goblets have cosmetic flaws and must be classified as seconds. a. Among six randomly selected goblets how likely is it that only one is a second b. Among six randomly selected goblets what is the probability that at least two are seconds c. If goblets are examined one by one what is the .