Exploring Poisson Probability Distribution

1. Chapter 6 Discrete Probability Distributions

2. Section 6.3 The Poisson Probability Distribution

3. Objectives • Determine whether a probability experiment follows a Poisson process • Compute probabilities of a Poisson random variable • Find the mean and standard deviation of a Poisson random variable

4. Objective 1 • Determine If a Probability Experiment Follows a Poisson Process

5. A random variable X, the number of successes in a fixed interval, follows a Poisson process provided the following conditions are met. • The probability of two or more successes in any sufficiently small subinterval is 0. • The probability of success is the same for any two intervals of equal length. • 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. 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.

7. Objective 2 • Compute Probabilities of a Poisson Random Variable

8. 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 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.

9. 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 10-gram sample of chocolate is(a) exactly three. Interpret the result.(b) fewer than three. Interpret the result.(c) at least three. Interpret the result.

10. (a) λ = 0.6; t = 10 (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

11. Objective 3 • Find the Mean and Standard Deviation of a Poisson Random Variable

12. 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 where t is the length of the interval.

13. 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

14. 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. • Determine the mean number of insect fragments in a 5 gram sample of chocolate. • What is the standard deviation?

15. EXAMPLE Mean and Standard Deviation of a Poisson Random Variable We would expect 3 insect fragments in a 5-gram sample of chocolate.

16. 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 α-particles emitted? Source: Rutherford, Sir Ernest; Chadwick, James; and Ellis, C.D.. Radiations from Radioactive Substances. London, Cambridge University Press, 1951, p. 172.