Normal Approximation to the Poisson Distribution in .NET

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Normal Approximation to the Poisson Distribution
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If X is a Poisson random variable with E1X 2 Z X
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and V1X 2
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, (4-13)
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is approximately a standard normal random variable. The approximation is good for 5
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EXAMPLE 4-20
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Assume that the number of asbestos particles in a squared meter of dust on a surface follows a Poisson distribution with a mean of 1000. If a squared meter of dust is analyzed, what is the probability that less than 950 particles are found This probability can be expressed exactly as
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1000 1000
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The computational dif culty is clear. The probability can be approximated as P1X x2 P aZ P1Z 1.582 0.057
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EXERCISES FOR SECTION 4-7
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4-61. Suppose that X is a binomial random variable with n 200 and p 0.4. (a) Approximate the probability that X is less than or equal to 70. (b) Approximate the probability that X is greater than 70 and less than 90. 4-62. Suppose that X is a binomial random variable with n 100 and p 0.1. (a) Compute the exact probability that X is less than 4. (b) Approximate the probability that X is less than 4 and compare to the result in part (a). (c) Approximate the probability that 8 X 12 .
CHAPTER 4 CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS
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4-63. The manufacturing of semiconductor chips produces 2% defective chips. Assume the chips are independent and that a lot contains 1000 chips. (a) Approximate the probability that more than 25 chips are defective. (b) Approximate the probability that between 20 and 30 chips are defective. 4-64. A supplier ships a lot of 1000 electrical connectors. A sample of 25 is selected at random, without replacement. Assume the lot contains 100 defective connectors. (a) Using a binomial approximation, what is the probability that there are no defective connectors in the sample (b) Use the normal approximation to answer the result in part (a). Is the approximation satisfactory (c) Redo parts (a) and (b) assuming the lot size is 500. Is the normal approximation to the probability that there are no defective connectors in the sample satisfactory in this case 4-65. An electronic office product contains 5000 electronic components. Assume that the probability that each component operates without failure during the useful life of the product is 0.999, and assume that the components fail independently. Approximate the probability that 10 or more of the original 5000 components fail during the useful life of the product. 4-66. Suppose that the number of asbestos particles in a sample of 1 squared centimeter of dust is a Poisson random variable with a mean of 1000. What is the probability that 10 squared centimeters of dust contains more than 10,000 particles 4-67. A corporate Web site contains errors on 50 of 1000 pages. If 100 pages are sampled randomly, without replace-
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ment, approximate the probability that at least 1 of the pages in error are in the sample. 4-68. Hits to a high-volume Web site are assumed to follow a Poisson distribution with a mean of 10,000 per day. Approximate each of the following: (a) The probability of more than 20,000 hits in a day (b) The probability of less than 9900 hits in a day (c) The value such that the probability that the number of hits in a day exceed the value is 0.01 4-69. Continuation of Exercise 4-68. (a) Approximate the expected number of days in a year (365 days) that exceed 10,200 hits. (b) Approximate the probability that over a year (365 days) more than 15 days each have more than 10,200 hits. 4-70. The percentage of people exposed to a bacteria who become ill is 20%. Assume that people are independent. Assume that 1000 people are exposed to the bacteria. Approximate each of the following: (a) The probability that more than 225 become ill (b) The probability that between 175 and 225 become ill (c) The value such that the probability that the number of people that become ill exceeds the value is 0.01 4-71. A high-volume printer produces minor print-quality errors on a test pattern of 1000 pages of text according to a Poisson distribution with a mean of 0.4 per page. (a) Why are the number of errors on each page independent random variables (b) What is the mean number of pages with errors (one or more) (c) Approximate the probability that more than 350 pages contain errors (one or more).
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