Math 160 - Cooley Intro to Statistics OCC

Section 5.4 – The Poisson Distribution

The Poisson Distribution

A discrete probability distribution that is often used to model the frequency with which a specified event occurs during a particular period of time.

In the picture above are several simultaneously portrayed Poisson distributions. When the rate of occurrence of some event is small, the range of likely possibilities will lie near the zero line. Meaning that when the rate is small, zero is a very likely number to get. As the rate becomes higher (as the occurrence of the thing we are watching becomes more common), the center of the curve moves toward the right, and eventually, somewhere around, zero occurrences actually become unlikely. This is how the Poisson world looks graphically.

For example, suppose you typically get 4 pieces of mail per day. That becomes your expectation, but there will be a certain spread: sometimes a little more, sometimes a little less, once in a while nothing at all. Given only the average rate, for a certain period of observation (such as pieces of mail per day, phone calls per hour, accidents at an intersection, babies born during a shift at a hospital wing), and assuming that the process, or mix of processes, that produce the event flow are essentially random, the Poisson Distribution will tell you how likely it is that you will get 3, or 5, or 11, or any other number, during one period of observation. That is, it predicts the degree of spread around a known average rate of occurrence. (The average or likeliest actual occurrence is the hump on each of the Poisson curves shown above). In certain instances, the Poisson Distribution can approximate the Binomial Distribution (the pattern of Heads and Tails in coin tosses). We may use the Poisson Distribution, with = np, as a computationally easier approximation to the Binomial Distribution when:

·  n ≥ 100, where n is the number of trials;

·  np ≤ 10, where p is the probability of success.

Poisson Probability Formula

Probabilities for a random variable X that has a Poisson distribution are given by the formula

x = 0, 1, 2, …,

where is a positive real number and . The random variable X is called a Poisson random variable and is said to have the Poisson distribution with parameter.

NOTE: A Poisson random variable has indefinitely many possible values–namely, all whole numbers. Consequently, we cannot display all the probabilities for a Poisson random variable in a probability distribution table.

Mean and Standard Deviation of a Poisson Random Variable

The mean and standard deviation of a Poisson random variable with parameter are

and ,

respectively.

To Approximate Binomial Probabilities by Using a Poisson Probability Formula

Step 1 – Find n, the number of trials, and p, the success probability.

Step 2 – Continue only if and .

Step 3 – Approximate the binomial probabilities by using the Poisson probability formula.

Note: In Step 3, we just substituted for np in the Poisson Probability Formula, since , when dealing with a binomial distribution.

J Exercises:

1) On an average Friday, a waitress gets no tips from 5 customers. Thus, let the random variable, X,

represent the number of “no tips” received on an average Friday. So, X has a Poisson distribution with

parameter .

a) Determine the probability that the waitress will get no tip from 7 customers this Friday?

b) Determine the probability that the waitress will get at most 3 customers that will not tip?

c) Determine the probability that the waitress will get at least 9 customers that will not tip?

d) Find the mean of the random variable X; that, is the mean number of “no tips” on an average

Friday.

e) Find the standard deviation of the random variable X; that, is the standard deviation number of

“no tips” on an average Friday.

2) According to the experts, the odds against a PGA golfer making a hole in one are 3708 to 1; that is, the probability is . Use the Poisson approximation to the binomial distribution to determine the probability that at least 4 of the 155 golfers playing the second round would get a hole in one on the sixth hole.

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