Section 5.5-1Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Lecture Slides

Elementary Statistics Twelfth Edition

and the Triola Statistics Series

by Mario F. Triola

Section 5.5-2Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Chapter 5Probability Distributions

5-1 Review and Preview

5-2 Probability Distributions

5-3 Binomial Probability Distributions

5-4 Parameters for Binomial Distributions

5-5 Poisson Probability Distributions

Section 5.5-3Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Key Concept

The Poisson distribution is another discrete probability distribution which is important because it is often used for describing the behavior of rare events (with small probabilities).

Section 5.5-4Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Poisson Distribution

The Poisson distribution is a discrete probability distribution that applies to occurrences of some event over a specified interval. The random variable x is the number of occurrences of the event in an interval. The interval can be time, distance, area, volume, or some similar unit.

Formula

( )!

where 2.71828

mean number of occurrences of the event over the interval

x eP x

xe

Section 5.5-5Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Requirements of thePoisson Distribution

The random variable x is the number of occurrences of an event over some interval.

The occurrences must be random.

The occurrences must be independent of each other.

The occurrences must be uniformly distributed over the interval being used.

Parameters

The mean is .

The standard deviation is

.

Section 5.5-6Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Differences from aBinomial Distribution

The Poisson distribution differs from the binomial distribution in these fundamental ways:

The binomial distribution is affected by the sample size n and the probability p, whereas the Poisson distribution is affected only by the mean .

In a binomial distribution the possible values of the random variable x are 0, 1, . . ., n, but a Poisson distribution has possible x values of 0, 1, 2, . . . , with no upper limit.

Section 5.5-7Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Example

For a recent period of 100 years, there were 530 Atlantic hurricanes. Assume the Poisson distribution is a suitable model.

a.Find μ, the mean number of hurricanes per year.

b.If P(x) is the probability of x hurricanes in a randomly selected year, find P(2).

Section 5.5-8Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Example

a. Find μ, the mean number of hurricanes per year.

b. If P(x) is the probability of x hurricanes in a randomly selected year, find P(2).

number of hurricanes 5305.3

number of years 100

5.325.3 2.718282 0.0701

! 2!

x eP

x

Section 5.5-9Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Poisson as an Approximation to the Binomial Distribution

Rule of Thumb to Use the Poisson to Approximate the Binomial

The Poisson distribution is sometimes used to approximate the binomial distribution when n is large and p is small.

100n

10np

Section 5.5-10Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Poisson as an Approximation to the Binomial Distribution -

Value for

If both of the following requirements are met,

then use the following formula to calculate ,

100n10np

n p

Section 5.5-11Copyright © 2014, 2012, 2010 Pearson Education, Inc.

ExampleIn the Maine Pick 4 game, you pay $0.50 to select a sequence of four digits, such as 2449.

If you play the game once every day, find the probability of winning at least once in a year with 365 days.

The chance of winning is

Then, we need μ: 1365 0.0365

10,000np

1

10,000p

Section 5.5-12Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Example - continuedBecause we want the probability of winning “at least” once, we will first find P(0).

There is a 0.9642 probability of no wins, so the probability of at least one win is:

0.036500.0365 2.718280 0.9642

0!P

1 0.9642 0.0358