The Poisson Distribution

We can use the Poisson distribution to estimate the

probability of arrivals at a car wash in one hour or the number of leaks in 100 miles of pipeline. Bell Labs

uses it to model the arrival of phone calls.

The Poisson Distribution

The Poisson distribution is defined by:

!)(

x

exf

x

Where f(x) is the probability of x occurrences in an interval

is the expected value or mean value of occurrences within an interval

e is the natural logarithm. e = 2.71828

Properties of the Poisson Distribution

1. The probability of occurrences is the same for any two intervals of equal length.

2. The occurrence or nonoccurrence of an event in one interval is independent of an occurrence on nonoccurrence of an event in any other interval

Example: Mercy Hospital

Poisson Probability Function

Patients arrive at the

emergency room of Mercy

Hospital at the average

rate of 6 per hour on

weekend evenings.

What is the

probability of 4 arrivals in

30 minutes on a weekend evening?

MERCYMERCY

Example: Mercy HospitalExample: Mercy Hospital

Poisson Probability FunctionPoisson Probability Function

4 33 (2.71828)(4) .1680

4!f

4 33 (2.71828)

(4) .16804!

f

MERCYMERCY

= 6/hour = 3/half-hour, = 6/hour = 3/half-hour, xx = 4 = 4

Using Excel to ComputeUsing Excel to ComputePoisson ProbabilitiesPoisson Probabilities

Formula WorksheetFormula Worksheet

A B

1 3 = Mean No. of Occurrences ( ) 2

3Number of Arrivals (x ) Probability f (x )

4 0 =POISSON(A4,$A$1,FALSE)5 1 =POISSON(A5,$A$1,FALSE)6 2 =POISSON(A6,$A$1,FALSE)7 3 =POISSON(A7,$A$1,FALSE)8 4 =POISSON(A8,$A$1,FALSE)9 5 =POISSON(A9,$A$1,FALSE)

10 6 =POISSON(A10,$A$1,FALSE)

… and so on … and so on

MERCYMERCY

Value WorksheetValue Worksheet

Using Excel to ComputeUsing Excel to ComputePoisson ProbabilitiesPoisson Probabilities

A B

1 3 = Mean No. of Occurrences ( ) 2

3Number of Arrivals (x ) Probability f (x )

4 0 0.04985 1 0.14946 2 0.22407 3 0.22408 4 0.16809 5 0.1008

10 6 0.0504

… and so on … and so on

MERCYMERCY

Poisson Distribution of ArrivalsPoisson Distribution of Arrivals

Example: Mercy HospitalExample: Mercy Hospital MERCYMERCY

Poisson Probabilities

0.00

0.05

0.10

0.15

0.20

0.25

0 1 2 3 4 5 6 7 8 9 10

Number of Arrivals in 30 Minutes

Pro

bab

ilit

y

actually, actually, the the

sequencesequencecontinues:continues:11, 12, …11, 12, …

Problem 31, p. 229

a. Write the appropriate Poisson distribution

b. What is the average number of occurrences in three time periods?

c. Write the appropriate Poisson function to determine the probability of x occurrences in three time periods.

d. Compute the probability of two occurrences in one time period.

e. Compute the probability of six occurrences in three time periods.

f. Compute the probability of five occurrences in two time periods.

Consider a Poisson probability distribution with an average number of occurrences of two per period.

Problem 31, p. 229

!

2)(

2

X

exf

x

6

!

6)(

6

X

exf

x

27067.2

5413.

!2

2)2(

22

e

f

(a)

(b)

(c)

(d)

Problem 31, p. 229

16062.!6

6)6(

66

e

f

15629.!5

4)5(

55

e

f(e)

(d)

Poisson Distribution for Ex. 39, p. 229

0

0.1

0.2

0.3

0 1 2 3 4 5 6 7

No. of Occurrences Per Interval

f(x

)

Problem 31, p. 229

The Hypergeometric Distribution

This is similar to the binominal distribution

except: (1) the trials are NOT independent; and (2) the probability of success (ρ) changes from trial to

trial.

Hypergeometric Distribution

Let r denote in the population size N labeled a success.

N – r is the number of elements in the population labeled failure.

The hypergeometric distribution is used to compute the probability that in a random

selection of n elements, selected without replacement, we obtain x elements labeled

success and N – x elements labeled failure.

Notice that the x successes must be pulled from the r

number of successes in the population and the n - x

failures must be drawn from a population of N – r failures

Hypergeometric Distribution

rx

n

N

xn

rN

x

r

xf

0 allfor )(

Where

n = the number of trials.

N = number of elements in the population

r = number of elements in the population labeled a success

Hypergeometric Distribution

rx

n

N

xn

rN

x

r

xf

0 allfor )(

Number of ways a sample of size n can be selected from a population of size N

Number of ways a sample of size x successes can be selected from a population of size r

Number of ways a sample of size n -x failures can be selected from a population of size N -r

Example: Neveready

Hypergeometric Probability DistributionBob Neveready has removed two

dead batteries from a flashlight andinadvertently mingled them withthe two good batteries he intendedas replacements. The four batteries look identical.

Bob now randomly selects two of the four batteries. What is the probability he selects the two good batteries?

ZAPZAP ZA

PZ

AP

ZAP ZAPZAPZAP

Example: Neveready Hypergeometric Probability Distribution

2 2 2! 2!

2 0 2!0! 0!2! 1( ) .167

4 4! 62 2!2!

r N r

x n xf x

N

n

2 2 2! 2!

2 0 2!0! 0!2! 1( ) .167

4 4! 62 2!2!

r N r

x n xf x

N

n

where:where: xx = 2 = number of = 2 = number of goodgood batteries selected batteries selected

nn = 2 = number of batteries selected = 2 = number of batteries selected NN = 4 = number of batteries in total = 4 = number of batteries in total rr = 2 = number of = 2 = number of goodgood batteries in total batteries in total

Using Excel to ComputeUsing Excel to ComputeHypergeometric ProbabilitiesHypergeometric Probabilities

Formula WorksheetFormula Worksheet

A B

1 2 Number of Successes (x ) 2 2 Number of Trials (n ) 3 2 Number of Elements in the Population Labeled Success (r ) 4 4 Number of Elements in the Population (N ) 5 6 f (x ) =HYPGEOMDIST(A1,A2,A3,A4)7

Value WorksheetValue Worksheet

Using Excel to ComputeUsing Excel to ComputeHypergeometric ProbabilitiesHypergeometric Probabilities

A B

1 2 Number of Successes (x ) 2 2 Number of Trials (n ) 3 2 Number of Elements in the Population Labeled Success (r ) 4 4 Number of Elements in the Population (N ) 5 6 f (x ) 0.16677