The Poisson Distribution Section 6.3

Objectives Compute probabilities for Poisson random variables Compute the mean, variance, and standard deviation of a Poisson random variable Use the Poisson distribution to approximate binomial probabilities

Compute probabilities for Poisson random variables (Hand Computation) Objective 1 Compute probabilities for Poisson random variables (Hand Computation)

Poisson Distribution Consider the situation where an advertising company has placed an ad on a website. The company’s managers are interested to know how many hits the website gets, on the average, during a certain time interval. Assume that between 2 P.M. and 3 P.M. there are an average of three hits per minute. If we let 𝑋 represent the number of hits that occur between 2 P.M. and 3 P.M., then under certain conditions 𝑋 will have a Poisson distribution. The Poisson distribution is a probability distribution that is used to describe certain events that occur in time or space.

Poisson Distribution Let 𝑋 be a random variable that represents the number of events that occur in a time interval of length 𝑡. Then 𝑋 will have the Poisson probability distribution if the following conditions are satisfied. The average rate at which events occur is the same at all times. The numbers of events that occur in non-overlapping time intervals are independent. For a very short interval of length 𝑡: It is essentially impossible for more than one event to occur within the time interval. The probability that one event occurs in the interval is approximately equal to λ𝑡, where λ is the average rate at which events occur.

Example – Poisson Distribution The number of hits on a website during a given time interval has a Poisson distribution. If there are an average of three hits per minute, and 𝑋 is the number of hits between 2:30 P.M. and 2:35 P.M., find λ and 𝑡. Solution: The average number of hits per minute is 3, so λ = 3 hits per minute. The length of the time interval between 2:30 and 2:35 is five minutes, so 𝑡 = 5 minutes.

Probabilities for the Poisson Distribution If the random variable 𝑋 has a Poisson distribution with a rate λ and time 𝑡, we can compute probabilities using the following distribution. For a Poisson random variable 𝑋 that represents the number of events that occur with average rate λ in a time interval of length 𝑡, the probability that 𝑥 events occur is 𝑃 𝑥 = 𝑒 −λ𝑡 λ𝑡 𝑥 𝑥! The number 𝑒 is a constant whose value is approximately 2.71828. The possible values for 𝑋 are 0, 1, 2, ⋯

Example – Poisson Probabilities The number of hits on a certain website follows a Poisson distribution with 𝜆 = 3 hits per minute. Let 𝑋 be the number of hits in a 2-minute period. Find the following probabilities: a) 𝑃(5) b) 𝑃(less than 3) c) 𝑃 more than 2 Solution: a) 𝑃 5 = 𝑒 −λ𝑡 λ𝑡 𝑥 𝑥! = 𝑒 −3∙2 3∙2 5 5! = 0.1606 b) 𝑃 less than 3 &=𝑃 0 +𝑃 1 +𝑃 2 &= 𝑒 −3∙2 3∙2 0 0! + 𝑒 −3∙2 3∙2 1 1! + 𝑒 −3∙2 3∙2 2 2! &=0.0025+0.0149+0.0446 &=0.0620 c) Note that the event “more than 2” is the complement of the event “less than 3.” Therefore, the 𝑃(more than 2) = 1 – 𝑃(less than 3) = 1 – 0.0620 = 0.9380

Application of Poisson Distribution The Poisson distribution can also be used to compute probabilities involving events that occur in a spatial region. In this situation, the quantity 𝑡 represents a spatial quantity such as length, area, or volume. Example: Yeast cells are suspended in a liquid medium at a concentration of 4 particles per milliliter. A volume of 2 milliliters is withdrawn. What is the probability that exactly 6 particles are contained in this volume? Solution: Let 𝑋 be the number of particles withdrawn. Then 𝑋 has a Poisson distribution. The rate is λ = 4 particles per milliliter, and the volume is 𝑡 = 2 milliliters. 𝑃(6) = 𝑒 −λ𝑡 λ𝑡 𝑥 𝑥! = 𝑒 −4·2 4·2 6 6! = 0.1221

Compute probabilities for Poisson random variables (TI-83 PLUS) Objective 1 Compute probabilities for Poisson random variables (TI-83 PLUS)

Poisson Distribution Consider the situation where an advertising company has placed an ad on a website. The company’s managers are interested to know how many hits the website gets, on the average, during a certain time interval. Assume that between 2 P.M. and 3 P.M., there are an average of three hits per minute. If we let 𝑋 represent the number of hits that occur between 2 P.M. and 3 P.M., then under certain conditions 𝑋 will have a Poisson distribution. The Poisson distribution is a probability distribution that is used to describe certain events that occur in time or space.

Poisson Distribution Let 𝑋 be a random variable that represents the number of events that occur in a time interval of length 𝑡. Then 𝑋 will have the Poisson probability distribution if the following conditions are satisfied. The average rate at which events occur is the same at all times. The numbers of events that occur in non-overlapping time intervals are independent. For a very short interval of length 𝑡: It is essentially impossible for more than one event to occur within the time interval. The probability that one event occurs in the interval is approximately equal to λ𝑡, where λ is the average rate at which events occur.

Example – Poisson Distribution The number of hits on a website during a given time interval has a Poisson distribution. If there are an average of three hits per minute, and 𝑋 is the number of hits between 2:30 P.M. and 2:35 P.M., find λ and 𝑡. Solution: The average number of hits per minute is 3, so λ = 3 hits per minute. The length of the time interval between 2:30 and 2:35 is five minutes, so 𝑡 = 5 minutes.

Probabilities for the Poisson Distribution If the random variable 𝑋 has a Poisson distribution with a rate λ and time 𝑡, we can compute probabilities using the following distribution. For a Poisson random variable 𝑋 that represents the number of events that occur with average rate λ in a time interval of length 𝑡, the probability that 𝑥 events occur is 𝑃 𝑥 = 𝑒 −λ𝑡 λ𝑡 𝑥 𝑥! The number 𝑒 is a constant whose value is approximately 2.71828. The possible values for 𝑋 are 0, 1, 2, ⋯

Poisson Probabilities on the TI-84 PLUS Similar to the binompdf and binomcdf commands, the poissonpdf and poissoncdf commands may be used to compute Poisson probabilities. The format of the poissonpdf command is poissonpdf(𝝀𝒕, 𝒙) and the format of the poissoncdf command is poissoncdf(𝝀𝒕, 𝒙).

Example – Poisson Probabilities The number of hits on a certain website follows a Poisson distribution with 𝜆 = 3 hits per minute. Let 𝑋 be the number of hits in a 2-minute period. Find the following probabilities: a) 𝑃(5) b) 𝑃(less than 3) c) 𝑃 more than 2 Solution: a) To find 𝑃(5), we use the poissonpdf command with input 𝝀𝒕 = (3)(2) = 6 and 𝑥 = 5. We find that 𝑃(5) is approximately 0.1606. b) To find 𝑃(less than 3), we use the poissoncdf command with input 𝝀𝒕 = (3)(2) = 6 and 𝑥 = 2, since the event “less than 3” is equivalent to the event “less than or equal to 2”. The result is approximately 0.0620. c) Note that the event “more than 2” is the complement of the event “less than 3.” Therefore, the 𝑃(more than 2) = 1 – 𝑃(less than 3) = 1 – 0.0620 = 0.9380

Application of Poisson Distribution The Poisson distribution can also be used to compute probabilities involving events that occur in a spatial region. In this situation, the quantity 𝑡 represents a spatial quantity such as length, area, or volume. Example: Yeast cells are suspended in a liquid medium at a concentration of 4 particles per milliliter. A volume of 2 milliliters is withdrawn. What is the probability that exactly 6 particles are contained in this volume? Solution: Let 𝑋 be the number of particles withdrawn. Then 𝑋 has a Poisson distribution. The rate is λ = 4 particles per milliliter, and the volume is 𝑡 = 2 milliliters. To find 𝑃(6), we use the poissonpdf command with input 𝝀𝒕 = (4)(2) =8 and 𝑥 =6.

Objective 2 Compute the mean, variance, and standard deviation of a Poisson random variable

Poisson – Mean, Variance, & Standard Deviation Let 𝑋 be a Poisson random variable with rate λ and interval length 𝑡. The mean of 𝑋 is 𝜇 𝑥 =λ𝑡. The variance of 𝑋 is 𝜎 𝑥 2 = λ𝑡. (Note: The variance is the same as mean) The standard deviation of 𝑋 is 𝜎 𝑥 = 𝜆𝑡 . Example: Yeast cells are suspended in a liquid medium at a concentration of 4 particles per milliliter. A volume of 2 milliliters is withdrawn. Let 𝑋 be the number of particles contained in this volume. Find the mean and standard deviation of 𝑋. Solution: 𝜆=4 and 𝑡=2, so the mean is 𝜇 𝑥 =𝜆𝑡=4∙2=8. The standard deviation is 𝜎 𝑥 = 𝜆𝑡 = 4∙2 =2.8284.

Use the Poisson distribution to approximate binomial probabilities Objective 3 Use the Poisson distribution to approximate binomial probabilities

Approximating the Binomial Distribution In the previous section, we learned how to compute probabilities for a binomial distribution. When the number of trials, 𝑛, is large, binomial probabilities can be hard to compute, even with technology. The reason is that when 𝑛 is large, 𝑛! is extremely large and will cause most forms of technology to overflow. When 𝑛 is large and 𝑝 is small, the Poisson distribution can be used to approximate the binomial distribution. The Poisson approximation may be used for any binomial probability for which 𝑛≥100 and 𝑝≤0.1. If 𝑛≥100 and 𝑝≤0.1, then the Binomial probabilities may be approximated by the Poisson distribution. Compute 𝑃(𝑥) by using the Poisson distribution with 𝑝 in place of 𝜆 and 𝑛 in place of 𝑡.

Example – Poisson Approximation A Nevada roulette wheel has 38 pockets, labeled 0, 00, and 1 through 36. If you bet on 00, the probability that you will win is 1 38 =0.0263. Assume you bet on 00 for 200 spins of the wheel. Find the probability that you win exactly six times. Solution: There are 200 trials, and each trial has success probability 0.0263. The number of successes has a binomial distribution with 𝑛=200 and 𝑝=0.0263. We compute 𝑃(6) by using the Poisson distribution with rate 𝜆=0.0263 and 𝑡=200. 𝑃 6 = 𝑒 −0.0263∙200 0.0263∙200 6 6! =0.153

You Should Know… How to tell if a random variable has a Poisson Distribution How to compute probabilities for Poisson random variables How to compute the mean, variance, and standard deviation of a Poisson random variable How to use the Poisson distribution to approximate binomial probabilities