QBM117Business Statistics

Probability Distributions

Poisson Distribution

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Objectives

• To introduce the Poisson distribution

• Learn how to recognize an experiment whose outcomes follow a Poisson probability distribution

• Learn how to calculate Poisson probabilities

• Find the mean and variance of a Poisson distribution

• Learn how to use Excel to find Poisson probabilities

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Poisson Distribution

• The binomial distribution can be used to calculate the probability of getting a specified number of successes for a given number of repeated trials.

• The Poisson distribution can be used to calculate the probability that there will be a specified number of occurrences within a unit of time or space.

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Poisson Experiment

A Poisson experiment possesses the following properties:1. The experiment consists of counting the number

of times a certain event occurs during a given unit of time or over a unit of space.

2. The probability that an event occurs in an interval is the same for all intervals of equal size, and is proportional to the size of the interval.

3. The number of events that occur in any interval is independent of the number of events that occur in any other interval.

4. The probability of more than one occurrence in a very small interval is close to zero.

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Example 1

A study was carried out to examine the number of emails received by employees of a large company. The study found that on average an employee receives 110 emails per week.

– The experiment consists of counting the number of emails received per week.

– If an average of 110 emails are received per week then an average of 110/5=22 will be received per day.

– The number of emails that arrive in a day is independent of the number of emails that arrive in any other day.

– Assume that emails arrive one at a time. 5

Poisson Random Variable

• The random variable in a Poisson experiment is the number of occurrences of an event, within a specified time or space.

• It is called the Poisson random variable.

• It is a discrete random variable as the possible values of the random variable can be listed: 0, 1, 2, …

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Examples of Poisson Random Variables

• The number of telephone calls received at a switchboard in a minute.

• The number of machine breakdowns during a night shift.

• The number of particles of a particular pollutant in a cubic meter of air emitted from a factory.

• The number of customers arriving at a bank in an hour.

• The number of flaws in a pane of glass.

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Poisson Distribution

• If X is a Poisson random variable, the probability distribution is given by

where is the average number of occurrences in a given time interval or region.

• Note that is a constant value of 2.71828… that can be found on your calculator.

( )!

xeP X x

x

e

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• Hence the probability of observing exactly x occurrences per unit of time or space is given by

( )!

xeP X x

x

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Example 2

A paint factory uses “Agent A” in the paint manufacturing process. There is an average of 3 particles of Agent A in a cubic meter of air emitted during the production process.

The number of Agent A particles has a Poisson distribution with mean 3 particles per cubic meter of air emitted from the factory.

Let X = the number of particles of Agent A in a cubic meter of air emitted from the factory.

= 3 particles per cubic meter.10

a. What is the probability that there will be 5 particles of Agent A in a cubic meter of air emitted from the factory?

3 53( 5)

5! 0.1008

eP X

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b. What is the probability that there will be no Agent A particles in a cubic meter of air emission?

3 03( 0)

0! 0.0498

eP X

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c. What is the probability that there will be less than 2 particles of agent A in a cubic meter of air emitted from the factory?

3 1

( 2) ( 0) ( 1)

3 0.0498

1! 0.0498 0.1494

0.1992

P X P X P X

e

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Example 3

We are interested in the number of arrivals at the drive-thru window of a fast-food restaurant during a 5 minute period. If we can assume that the probability of a car arriving is the same for any two periods of equal length and that the number of arrivals in any period is independent of the number of arrivals in any other period, then the number of arrivals can be modelled as a Poisson random variable. Suppose that we are only interested in the busy lunch hour period and these assumptions are satisfied. The average number of cars arriving in a 5 minute period is 4.

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Let X = the number of cars arriving in a 5 minute period.

= 4 cars per 5 minute period

X has a Poisson distribution with a mean of 4 cars per 5 minute period.

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a. What is the probability of exactly 5 cars arrivals during a 5 minute period?

1563.0 !5

4)5(

54

e

XP

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b. What is the probability of 5 arrivals during a 10 minute period?

We are now interested in a 10 minute period rather than a 5 minute period.

= 4 cars per 5 minute period, is equivalent to

= 8 cars per 10 minute period.

0916.0 !5

8)5(

58

e

XP

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Poisson Tables

• An alternative to calculating Poisson probabilities using the formula is to use Table 2 of Appendix C of the text.

• The probabilities given in this table are cumulative probabilities.

)(

)(...)1()0()(

0

k

x

xXP

kXPXPXPkXP

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Example 4

The number of accidents that occur at a busy intersection is Poisson distributed with a mean of 3.5 per week. Find the probability of the following events:

a. Less than three accidents in a week

b. Five or more accidents in a week

c. No accidents today

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Let X = the number of accidents per week

= 3.5 accidents per week

a. The probability that there less than three accidents in a week is

321.0

)2()3(

XPXP

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b. The probability that there are five or more accidents in a week is

275.0

725.01

)4(1)5(

XPXP

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c. We want to find the probability that there are no accidents today.

The average number of accidents per week is 3.5.

Therefore the average number of accidents per day is 3.5/7=0.5.

Hence= 0.5 accidents per day

The probability that there are no accidents today is

607.0)1( XP

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Expected Value and Variance of Poisson Random Variables

• If X is a Poisson random variable, the mean and variance of X are

where is the average number of occurrences in a given interval of time or space.

)(

)(

XV

XE

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Example 4 revisited

The number of accidents that occur at a busy intersection is Poisson distributed with a mean of 3.5 per week.

a. What is the expected number of accidents per week?

b. What is the variance of the number of accidents per week?

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The expected number of accidents per week:

The variance of the number of accidents per week:

5.3)( XE

5.3)( XV

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Calculating Binomial and Poisson Probabilities in Excel

• There are instructions on page 204 of the text (pg 202 abridged).

• You will be asked to calculate binomial and Poisson probabilities in Excel in Tutorial 5.

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Reading for next lecture

• Chapter 5 Section 5.6 – 5.7

Exercises

• 5.38• 5.45• 5.75

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