m16 poisson distribution 1 department of ism, university of alabama, 1995-2003 lesson objectives ...

M16 Poisson Distribution 1 Department of ISM, University of Alabama, 1995-2003

Lesson Objectives

Learn when to use the Poisson distribution.

Learn how to calculate probabilitiesfor the Poisson using the formulaand the two tables in the book.

Understand the inverse relationship between the Poisson and exponential.


A data distribution used to model the count of the number of

occurrences of some event over a specified span of

time, space, or distance.

Poisson Distribution:

Quantitative, discrete

Examples of Poisson Variables

Number of tornados striking Alabama per week during next spring (March, April, May).

Number of flaws in the next 100 sq. yd. of fabric produced at a textile mill

Number of potholes per mile on city streets.

Number of incoming calls to a 911 switchboard during a one-day period

Number of customers arriving at a store in a one-hour period.


Poisson Distribution

• Fixed span of time, space, or distance .

• X = count of number of occurrences of event

• Possible values of a Poisson variable:

0, 1, 2, 3, … , (whole numbers)

• One parameter: the average number of occurrences in the specified span of time, space, or distance

• Notation: X ~ Poisson( = Mean )


Additional Poisson Assumptions:

The number of occurrences in one interval is independent of the number in any other non-overlapping

interval.

The average number of occurrences in an interval is proportional to the size of the interval.

Two or more events can’t occur at the same time or place.


Are these Poisson variables?

• Number of children in a family?

• Number of cars passing through an intersection in 5 minutes?

• Number of hits on your Web site in 24 hours?

• Number of birdies in a round of golf?


The probability that an event will occur exactly x times in a given span of time, space, or distance is:

P( ) for = 0, 1, 2, ,!

xex x

x

The Poisson distribution:


Poi = is population std. dev.

If a population of X values follows a Poisson( ) distribution, then:

Poi = is population mean

Parameter for the Poisson

See formula sheet


Example: Phone calls arrive at a switchboard at an average rate of 2.0 calls per minute. If the number of calls in any time interval follows the Poisson distribution, then

X = number of phone calls in a given minute.X ~ Poisson ( =

Y = number of phone calls in a given hour.Y ~ Poisson ( =

W = number of phone calls in a 15 seconds.W ~ Poisson ( =


!

xex

a. Find the probability of exactly five calls in the next three minutes?

Y = number of calls in next three minutes

7776 • .002479120

Y ~ Poisson ( =

P(Y = 5) =

=

=

See page 902,Table A.3, = 6.0, k = 5.

b. Find the probability of at least two phone calls in the next three minutes?

P(x 2) = 1.0 – [P(x = 0) + P(x = 1)]

60 • e–6

0!1 • .002479

1

x: 0 1 2 3 4 5 6

61 • e–6

1!

= .982647

6 • .0024791

= 6.0wantdon’t want

P(x = 0) = = = .002479

P(x = 1) = = = .014874

P(x 2) = 1.0 – [.002479 + .014874)]

p. 902, Table A.3, = 6.0, k = 0, 1.

p. 902, Table A.4, = 6.0, k = 1, for P(x 1)


Table A3, page 901-904; Individual

Table A4, page 901-904; Cumulative

= 4.1

P(X = 3) = _________

= 4.1

P(X 3) =

= _________

P(X=0) + P(X=1) + P(X=2) + P(X=3)

Poisson Tables


Simulation: The relationship between the Poisson and Exponential distributions.

Situation: The time at which a web site receives a “hit” is randomly generated over a 96 minute period.

Y = the “time between hits.”

Y ~ Exponential( = 4 min./ hit)


1.377 1.377 6.558 7.934 0.692 8.626 0.954 9.580 1.336 10.916 2.082 12.998 5.589 18.58711.023 29.61018.632 48.243 0.235 48.478 0.098 48.576 0.172 48.748 4.100 52.848 4.995 57.843 3.995 61.837 1.982 63.819 7.274 71.093 . . . . . .

X = 25 hits Y = 3.84 min./hitsY = 4.34 min./hit

X = 25 hits/ 96 min. X = .260 hits/ 1 min.

Y = Time IntervalsY ~ Exp( = 4.0 min/hit)

X = Count of “Hits”X ~ Poi( = .25 hits/min.)

Elapsed Time

Time Intervals

1 1 3 1 1 0 0 1 0 0 0 0 4 1 1 2 0 1 1 1 0 3 0 3

W = 1.042 hits/ 4 min.Histogram

0

2

4

6

8

10

12

0 1 2 3 4 5 6 MoreCounts per 4 minutes

Fre

qu

ency

SampleExponentialDistribution

SamplePoisson

Distribution


1.625 1.625 5.811 7.436 4.258 11.694 1.248 12.943 1.869 14.812 6.327 21.139 0.107 21.246 2.202 23.448 5.726 29.174 2.439 31.613 1.961 33.574 4.172 37.746 1.966 39.712 0.968 40.680 0.960 41.640 0.315 41.956 2.086 44.041 . . . . . .



X = Count of Occ.X ~ Poi( = .25 hits/min.)

Elapsed Time

Time Intervals

W = 1.208 hits/ 4 min.Sample

ExponentialDistribution

1 1 1 2 0 3 0 2 1 2 3 1 1 1 1 0 1 3 1 1 1 0 2 0

0

2

4

6

8

10

12

14


Fre

qu

ency

SamplePoisson

Distribution



1.625 1.625 0.138 1.76317.340 19.099 1.986 21.085 0.692 21.778 2.144 23.921 3.441 27.363 3.906 31.268 1.188 32.457 4.732 37.189 1.892 39.081 1.249 40.330 3.409 43.73912.777 56.516 3.596 60.112 0.119 60.231 2.718 62.949 . . . . . .




X = Count of Occ.X ~ Poi( = .25 hits/min.)

Elapsed Time

Time Intervals

W = 1.042 hits/ 4 min.

SampleExponentialDistribution

2 0 0 0 1 3 1 1 1 2 2 0 0 0 1 4 0 1 3 0 0 2 0 1

0

2

4

6

8

10

12


Fre

qu

ency

SamplePoisson

Distribution


In symbols:

X = number of arrivals ~ Poisson( )

Y = time between arrivals ~ exponential(1/ )

If the number of “occurrences” in a fixed interval has a Poisson distribution, then the times between “occurrences” have an exponential distribution.

The mean of the The mean of the exponentialexponential is the is the inverseinverse of the mean of the of the mean of the PoissonPoissonThe mean of the The mean of the exponentialexponential is the is the inverseinverse of the mean of the of the mean of the PoissonPoisson

Relationship between Poisson and Exponential


The End

m16 poisson distribution 1 department of ism, university of alabama, 1995-2003 lesson objectives ...

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