Transcript Ch6-Sec6.3
Chapter 6 Discrete Probability Distributions Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Section 6.3 The Poisson Probability Distribution Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Objectives 1. Determine whether a probability experiment follows a Poisson process 2. Compute probabilities of a Poisson random variable 3. Find the mean and standard deviation of a Poisson random variable 6-3 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Objective 1 • Determine If a Probability Experiment Follows a Poisson Process 6-4 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. A random variable X, the number of successes in a fixed interval, follows a Poisson process provided the following conditions are met. 1. The probability of two or more successes in any sufficiently small subinterval is 0. 2. The probability of success is the same for any two intervals of equal length. 3. The number of successes in any interval is independent of the number of successes in any other interval provided the intervals are not overlapping. 6-5 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. EXAMPLE Illustrating a Poisson Process The Food and Drug Administration sets a Food Defect Action Level (FDAL) for various foreign substances that inevitably end up in the food we eat and liquids we drink. For example, the FDAL level for insect filth in chocolate is 0.6 insect fragments (larvae, eggs, body parts, and so on) per 1 gram. 6-6 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Objective 2 • Compute Probabilities of a Poisson Random Variable 6-7 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Poisson Probability Distribution Function If X is the number of successes in an interval of fixed length t, then the probability of obtaining x successes in the interval is P x t x x! e t x 0,1, 2, 3,... where λ (the Greek letter lambda) represents the average number of occurrences of the event in some interval of length 1 and e ≈ 2.71828. 6-8 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. EXAMPLE Illustrating a Poisson Process The Food and Drug Administration sets a Food Defect Action Level (FDAL) for various foreign substances that inevitably end up in the food we eat and liquids we drink. For example, the FDAL level for insect filth in chocolate is 0.6 insect fragments (larvae, eggs, body parts, and so on) per 1 gram. Suppose that a chocolate bar has 0.6 insect fragments per gram. Compute the probability that the number of insect fragments in a 10gram sample of chocolate is (a) exactly three. Interpret the result. (b) fewer than three. Interpret the result. (c) at least three. Interpret the result. 6-9 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. (a) λ = 0.6; t = 10 (0.6 10)3 0.6(10) P(3) e 3! 0.0892 (b) P(X < 3) = P(X < 2) = P(0) + P(1) + P(2) = 0.0620 (c) P(X > 3) = 1 – P(X < 2) = 1 – 0.0620 = 0.938 6-10 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Objective 3 • Find the Mean and Standard Deviation of a Poisson Random Variable 6-11 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Mean and Standard Deviation of a Poisson Random Variable A random variable X that follows a Poisson process with parameter λ has mean (or expected value) and standard deviation given by the formulas X t and X t X where t is the length of the interval. 6-12 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Poisson Probability Distribution Function If X is the number of successes in an interval of fixed length and X follows a Poisson process with mean μ, the probability distribution function for X is P x 6-13 x x! e x 0,1, 2, 3,... Copyright © 2013, 2010 and 2007 Pearson Education, Inc. EXAMPLE Mean and Standard Deviation of a Poisson Random Variable The Food and Drug Administration sets a Food Defect Action Level (FDAL) for various foreign substances that inevitably end up in the food we eat and liquids we drink. For example, the FDAL level for insect filth in chocolate is 0.6 insect fragments (larvae, eggs, body parts, and so on) per 1 gram. (a)Determine the mean number of insect fragments in a 5 gram sample of chocolate. (b) What is the standard deviation? 6-14 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. EXAMPLE Poisson Mean and Standard Deviation of a Random Variable (a) X t (0.6)(5) 3 We would expect 3 insect fragments in a 5-gram sample of chocolate. (b) X t (0.6)(5) 3 6-15 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. EXAMPLE A Poisson Process? In 1910, Ernest Rutherford and Hans Geiger recorded the number of α-particles emitted from a polonium source in eighth-minute (7.5 second) intervals. The results are reported in the table to the right. Does a Poisson probability function accurately describe the number of αparticles emitted? Source: Rutherford, Sir Ernest; Chadwick, James; and Ellis, C.D.. Radiations from Radioactive Substances. London, Cambridge University Press, 1951, p. 172. 6-16 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. 6-17 Copyright © 2013, 2010 and 2007 Pearson Education, Inc.