Slides Prepared by JOHN S. LOUCKS St. Edward’s University

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© 2006 Thomson/South-Western© 2006 Thomson/South-Western

Slides Prepared bySlides Prepared by

JOHN S. LOUCKSJOHN S. LOUCKSSt. Edward’s UniversitySt. Edward’s University

Slides Prepared bySlides Prepared by

JOHN S. LOUCKSJOHN S. LOUCKSSt. Edward’s UniversitySt. Edward’s University

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Chapter 6Chapter 6 Continuous Probability Distributions Continuous Probability Distributions

Uniform Probability DistributionUniform Probability Distribution Normal Probability DistributionNormal Probability Distribution Exponential Probability DistributionExponential Probability Distribution

f (x)f (x)

x x

UniformUniform

xx

f f ((xx)) NormalNormal

xx

f (x)f (x) ExponentialExponential

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Continuous Probability DistributionsContinuous Probability Distributions

A A continuous random variablecontinuous random variable can assume any can assume any value in an interval on the real line or in a value in an interval on the real line or in a collection of intervals.collection of intervals.

It is not possible to talk about the probability of the It is not possible to talk about the probability of the random variable assuming a particular value.random variable assuming a particular value.

Instead, we talk about the probability of the random Instead, we talk about the probability of the random variable assuming a value within a given interval.variable assuming a value within a given interval.

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Continuous Probability DistributionsContinuous Probability Distributions

The probability of the random variable The probability of the random variable assuming a value within some given interval assuming a value within some given interval from from xx11 to to xx22 is defined to be the is defined to be the area under area under the graphthe graph of the of the probability density functionprobability density function betweenbetween x x11 andand x x22..

f (x)f (x)

x x

UniformUniform

xx11 xx11 xx22 xx22

xx

f f ((xx)) NormalNormal

xx11 xx11 xx22 xx22

xx11 xx11 xx22 xx22

ExponentialExponential

xx

f (x)f (x)

xx11

xx11

xx22 xx22

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Uniform Probability DistributionUniform Probability Distribution

where: where: aa = smallest value the variable can assume = smallest value the variable can assume

bb = largest value the variable can assume = largest value the variable can assume

f f ((xx) = 1/() = 1/(bb – – aa) for ) for aa << xx << bb = 0 elsewhere= 0 elsewhere f f ((xx) = 1/() = 1/(bb – – aa) for ) for aa << xx << bb = 0 elsewhere= 0 elsewhere

A random variable is A random variable is uniformly distributeduniformly distributed whenever the probability is proportional to the whenever the probability is proportional to the interval’s length. interval’s length.

The The uniform probability density functionuniform probability density function is: is:

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Var(Var(xx) = () = (bb - - aa))22/12/12Var(Var(xx) = () = (bb - - aa))22/12/12

E(E(xx) = () = (aa + + bb)/2)/2E(E(xx) = () = (aa + + bb)/2)/2


Expected Value of Expected Value of xx

Variance of Variance of xx

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Example: Slater's BuffetExample: Slater's BuffetSlater customers are Slater customers are

chargedchargedfor the amount of salad they take. for the amount of salad they take. Sampling suggests that theSampling suggests that theamount of salad taken is amount of salad taken is uniformly distributeduniformly distributedbetween 5 ounces and 15 ounces.between 5 ounces and 15 ounces.

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Uniform Probability Density FunctionUniform Probability Density Function

ff((xx) = 1/10 for 5 ) = 1/10 for 5 << xx << 15 15

= 0 elsewhere= 0 elsewhere

ff((xx) = 1/10 for 5 ) = 1/10 for 5 << xx << 15 15

= 0 elsewhere= 0 elsewhere

where:where:

xx = salad plate filling weight = salad plate filling weight


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Expected Value of Expected Value of xx

Variance of Variance of xx

E(E(xx) = () = (aa + + bb)/2)/2

= (5 + 15)/2= (5 + 15)/2

= 10= 10

E(E(xx) = () = (aa + + bb)/2)/2

= (5 + 15)/2= (5 + 15)/2

= 10= 10

Var(Var(xx) = () = (bb - - aa))22/12/12

= (15 – 5)= (15 – 5)22/12/12

= 8.33= 8.33

Var(Var(xx) = () = (bb - - aa))22/12/12

= (15 – 5)= (15 – 5)22/12/12

= 8.33= 8.33


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Uniform Probability DistributionUniform Probability Distributionfor Salad Plate Filling Weightfor Salad Plate Filling Weight

f(x)f(x)

x x55 1010 1515

1/101/10

Salad Weight (oz.)Salad Weight (oz.)


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f(x)f(x)

x x55 1010 1515

1/101/10

Salad Weight (oz.)Salad Weight (oz.)

P(12 < x < 15) = 1/10(3) = .3P(12 < x < 15) = 1/10(3) = .3

What is the probability that a customerWhat is the probability that a customer

will take between 12 and 15 ounces of will take between 12 and 15 ounces of salad?salad?

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Normal Probability DistributionNormal Probability Distribution

The The normal probability distributionnormal probability distribution is the most is the most important distribution for describing a important distribution for describing a continuous random variable.continuous random variable.

It is widely used in statistical inference.It is widely used in statistical inference.

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HeightsHeightsof peopleof peopleHeightsHeights

of peopleof people


It has been used in a wide variety of It has been used in a wide variety of applications:applications:

ScientificScientific measurementsmeasurements

ScientificScientific measurementsmeasurements

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AmountsAmounts

of rainfallof rainfall

AmountsAmounts

of rainfallof rainfall


It has been used in a wide variety of It has been used in a wide variety of applications:applications:

TestTest scoresscoresTestTest

scoresscores

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Normal Probability Density FunctionNormal Probability Density Function

2 2( ) / 21( )

2xf x e

2 2( ) / 21( )

2xf x e

= mean= mean

= standard deviation= standard deviation

= 3.14159= 3.14159

ee = 2.71828 = 2.71828

where:where:

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The distribution is The distribution is symmetricsymmetric; its skewness; its skewness measure is zero.measure is zero. The distribution is The distribution is symmetricsymmetric; its skewness; its skewness measure is zero.measure is zero.


CharacteristicsCharacteristics

xx

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The entire family of normal probabilityThe entire family of normal probability distributions is defined by itsdistributions is defined by its meanmean and its and its standard deviationstandard deviation . .

The entire family of normal probabilityThe entire family of normal probability distributions is defined by itsdistributions is defined by its meanmean and its and its standard deviationstandard deviation . .



Standard Deviation Standard Deviation

Mean Mean xx

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The The highest pointhighest point on the normal curve is at the on the normal curve is at the meanmean, which is also the , which is also the medianmedian and and modemode.. The The highest pointhighest point on the normal curve is at the on the normal curve is at the meanmean, which is also the , which is also the medianmedian and and modemode..



xx

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-10-10 00 2020

The mean can be any numerical value: negative,The mean can be any numerical value: negative, zero, or positive.zero, or positive. The mean can be any numerical value: negative,The mean can be any numerical value: negative, zero, or positive.zero, or positive.

xx

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= 15= 15

= 25= 25

The standard deviation determines the width of theThe standard deviation determines the width of thecurve: larger values result in wider, flatter curves.curve: larger values result in wider, flatter curves.The standard deviation determines the width of theThe standard deviation determines the width of thecurve: larger values result in wider, flatter curves.curve: larger values result in wider, flatter curves.

xx

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Probabilities for the normal random variable areProbabilities for the normal random variable are given by given by areas under the curveareas under the curve. The total area. The total area under the curve is 1 (.5 to the left of the mean andunder the curve is 1 (.5 to the left of the mean and .5 to the right)..5 to the right).

Probabilities for the normal random variable areProbabilities for the normal random variable are given by given by areas under the curveareas under the curve. The total area. The total area under the curve is 1 (.5 to the left of the mean andunder the curve is 1 (.5 to the left of the mean and .5 to the right)..5 to the right).



.5.5 .5.5

xx

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of values of a normal random variableof values of a normal random variable are within of its mean.are within of its mean.

of values of a normal random variableof values of a normal random variable are within of its mean.are within of its mean.68.26%68.26%68.26%68.26%

+/- 1 standard deviation+/- 1 standard deviation+/- 1 standard deviation+/- 1 standard deviation



+/- 2 standard deviations+/- 2 standard deviations+/- 2 standard deviations+/- 2 standard deviations



+/- 3 standard deviations+/- 3 standard deviations+/- 3 standard deviations+/- 3 standard deviations

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xx – – 33 – – 11

– – 22 + 1+ 1

+ 2+ 2 + 3+ 3

68.26%68.26%95.44%95.44%99.72%99.72%

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Standard Normal Probability DistributionStandard Normal Probability Distribution

A random variable having a normal distributionA random variable having a normal distribution with a mean of 0 and a standard deviation of 1 iswith a mean of 0 and a standard deviation of 1 is said to have a said to have a standard normal probabilitystandard normal probability distributiondistribution..

A random variable having a normal distributionA random variable having a normal distribution with a mean of 0 and a standard deviation of 1 iswith a mean of 0 and a standard deviation of 1 is said to have a said to have a standard normal probabilitystandard normal probability distributiondistribution..

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00zz

The letter The letter z z is used to designate the standardis used to designate the standard normal random variable.normal random variable. The letter The letter z z is used to designate the standardis used to designate the standard normal random variable.normal random variable.


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Converting to the Standard Normal Converting to the Standard Normal DistributionDistribution


zx

zx

We can think of We can think of zz as a measure of the number of as a measure of the number ofstandard deviations standard deviations xx is from is from ..

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Standard Normal Density FunctionStandard Normal Density Function

2 / 21( )

2zf x e

2 / 21( )

2zf x e

z z = ( = (xx – – )/)/ = 3.14159= 3.14159

ee = 2.71828 = 2.71828

where:where:

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Example: Pep ZoneExample: Pep Zone

Pep Zone sells auto parts and supplies Pep Zone sells auto parts and supplies includingincluding

a popular multi-grade motor oil. When thea popular multi-grade motor oil. When the

stock of this oil drops to 20 gallons, astock of this oil drops to 20 gallons, a

replenishment order is placed.replenishment order is placed. PepZone5w-20Motor Oil

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The store manager is concerned that sales The store manager is concerned that sales are beingare being

lost due to stockouts while waiting for an order.lost due to stockouts while waiting for an order.

It has been determined that demand duringIt has been determined that demand during

replenishment lead-time is normallyreplenishment lead-time is normally

distributed with a mean of 15 gallons anddistributed with a mean of 15 gallons and

a standard deviation of 6 gallons. a standard deviation of 6 gallons.

The manager would like to know theThe manager would like to know the

probability of a stockout, probability of a stockout, PP((xx > 20). > 20).


PepZone5w-20Motor Oil

Example: Pep ZoneExample: Pep Zone

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zz = ( = (xx - - )/)/ = (20 - 15)/6= (20 - 15)/6 = .83= .83

zz = ( = (xx - - )/)/ = (20 - 15)/6= (20 - 15)/6 = .83= .83

Solving for the Stockout ProbabilitySolving for the Stockout Probability

Step 1: Convert Step 1: Convert xx to the standard normal distribution. to the standard normal distribution.Step 1: Convert Step 1: Convert xx to the standard normal distribution. to the standard normal distribution.

PepZone5w-20

Motor Oil

Step 2: Find the area under the standard normalStep 2: Find the area under the standard normal curve to the left of curve to the left of zz = .83. = .83.Step 2: Find the area under the standard normalStep 2: Find the area under the standard normal curve to the left of curve to the left of zz = .83. = .83.

see next slidesee next slide see next slidesee next slide


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Cumulative Probability Table for Cumulative Probability Table for the Standard Normal Distributionthe Standard Normal Distribution

z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09

. . . . . . . . . . .

.5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224

.6 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2517 .2549

.7 .2580 .2611 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852

.8 .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3133

.9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389

. . . . . . . . . . .

z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09

. . . . . . . . . . .

.5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224

.6 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2517 .2549

.7 .2580 .2611 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852

.8 .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3133

.9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389

. . . . . . . . . . .

PepZone5w-20

Motor Oil

PP(.00 (.00 << zz << .83) .83)


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PP((z z > .83) = .5 – > .83) = .5 – PP(.00 (.00 << zz << .83) .83) = .5 = .5 .2967 .2967

= .2033= .2033

PP((z z > .83) = .5 – > .83) = .5 – PP(.00 (.00 << zz << .83) .83) = .5 = .5 .2967 .2967

= .2033= .2033


Step 3: Compute the area under the standard normalStep 3: Compute the area under the standard normal curve to the right of curve to the right of zz = .83. = .83.Step 3: Compute the area under the standard normalStep 3: Compute the area under the standard normal curve to the right of curve to the right of zz = .83. = .83.

PepZone5w-20

Motor Oil

ProbabilityProbability of a of a

stockoutstockoutPP((xx > > 20)20)


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00 .83.83

Area = .2967Area = .2967Area = .5 Area = .5 .2967 .2967

= .2033= .2033

zz

PepZone5w-20

Motor Oil


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If the manager of Pep Zone wants the If the manager of Pep Zone wants the probability of a stockout to be no more probability of a stockout to be no more than .05, what should the reorder point be?than .05, what should the reorder point be?

PepZone5w-20

Motor Oil


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Solving for the Reorder PointSolving for the Reorder Point

PepZone5w-20

Motor Oil

00

Area = .4500Area = .4500

Area = .0500Area = .0500

zzzz.05.05


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PepZone5w-20

Motor Oil

Step 1: Find the Step 1: Find the zz-value that cuts off an area of .05-value that cuts off an area of .05 in the right tail of the standard normalin the right tail of the standard normal distribution.distribution.

Step 1: Find the Step 1: Find the zz-value that cuts off an area of .05-value that cuts off an area of .05 in the right tail of the standard normalin the right tail of the standard normal distribution.distribution.

z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09

. . . . . . . . . . .

1.5 .4332 .4345 .4357 .4370 .4382 .4394 .4406 .4418 .4429 .4441

1.6 .4452 .4463 .4474 .4484 .4495 .4505 .4515 .4525 .4535 .4545

1.7 .4554 .4564 .4573 .4582 .4591 .4599 .4608 .4616 .4625 .4633

1.8 .4641 .4649 .4656 .4664 .4671 .4678 .4686 .4693 .4699 .4706

1.9 .4713 .4719 .4726 .4732 .4738 .4744 .4750 .4756 .4761 .4767

. . . . . . . . . . .

z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09

. . . . . . . . . . .

1.5 .4332 .4345 .4357 .4370 .4382 .4394 .4406 .4418 .4429 .4441

1.6 .4452 .4463 .4474 .4484 .4495 .4505 .4515 .4525 .4535 .4545

1.7 .4554 .4564 .4573 .4582 .4591 .4599 .4608 .4616 .4625 .4633

1.8 .4641 .4649 .4656 .4664 .4671 .4678 .4686 .4693 .4699 .4706

1.9 .4713 .4719 .4726 .4732 .4738 .4744 .4750 .4756 .4761 .4767

. . . . . . . . . . .We look up .5 minusWe look up .5 minusthe tail area (.5 - .05 the tail area (.5 - .05

= .45)= .45)


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PepZone5w-20

Motor Oil

Step 2: Convert Step 2: Convert zz.05.05 to the corresponding value of to the corresponding value of xx..Step 2: Convert Step 2: Convert zz.05.05 to the corresponding value of to the corresponding value of xx..

xx = = + + zz.05.05

= 15 + 1.645(6)= 15 + 1.645(6)

= 24.87 or 25= 24.87 or 25

xx = = + + zz.05.05

= 15 + 1.645(6)= 15 + 1.645(6)

= 24.87 or 25= 24.87 or 25

A reorder point of 25 gallons will place the probabilityA reorder point of 25 gallons will place the probability of a stockout during leadtime at (slightly less than) .05.of a stockout during leadtime at (slightly less than) .05.


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PepZone5w-20

Motor Oil

By raising the reorder point from 20 gallons to By raising the reorder point from 20 gallons to 25 gallons on hand, the probability of a stockout25 gallons on hand, the probability of a stockoutdecreases from about .20 to .05.decreases from about .20 to .05. This is a significant decrease in the chance that PepThis is a significant decrease in the chance that PepZone will be out of stock and unable to meet aZone will be out of stock and unable to meet acustomer’s desire to make a purchase.customer’s desire to make a purchase.


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Exponential Probability DistributionExponential Probability Distribution

The exponential probability distribution is The exponential probability distribution is useful in describing the time it takes to useful in describing the time it takes to complete a task.complete a task.

The exponential random variables can be used The exponential random variables can be used to describe:to describe:

Time betweenTime betweenvehicle arrivalsvehicle arrivalsat a toll boothat a toll booth

Time betweenTime betweenvehicle arrivalsvehicle arrivalsat a toll boothat a toll booth

Time requiredTime requiredto completeto complete

a questionnairea questionnaire

Time requiredTime requiredto completeto complete

a questionnairea questionnaire

Distance betweenDistance betweenmajor defectsmajor defectsin a highwayin a highway

Distance betweenDistance betweenmajor defectsmajor defectsin a highwayin a highway

SLOW

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Density FunctionDensity Function


where: where: = mean = mean

ee = 2.71828 = 2.71828

f x e x( ) / 1

f x e x( ) / 1

for for xx >> 0, 0, > 0 > 0

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Cumulative ProbabilitiesCumulative Probabilities


P x x e x( ) / 0 1 o P x x e x( ) / 0 1 o

where:where:

xx00 = some specific value of = some specific value of xx

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Example: Al’s Full-Service PumpExample: Al’s Full-Service Pump

The time between arrivals of carsThe time between arrivals of carsat Al’s full-service gas pump followsat Al’s full-service gas pump followsan exponential probability distributionan exponential probability distributionwith a mean time between arrivals of with a mean time between arrivals of 3 minutes. Al would like to know the3 minutes. Al would like to know theprobability that the time between two successiveprobability that the time between two successivearrivals will be 2 minutes or less.arrivals will be 2 minutes or less.

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xx

f(x)f(x)

.1.1

.3.3

.4.4

.2.2

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10Time Between Successive Arrivals (mins.)Time Between Successive Arrivals (mins.)


PP((xx << 2) = 1 - 2.71828 2) = 1 - 2.71828-2/3-2/3 = 1 - .5134 = .4866 = 1 - .5134 = .4866 PP((xx << 2) = 1 - 2.71828 2) = 1 - 2.71828-2/3-2/3 = 1 - .5134 = .4866 = 1 - .5134 = .4866

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A property of the exponential distribution is thatA property of the exponential distribution is that the mean, the mean, , and standard deviation, , and standard deviation, , are equal., are equal. A property of the exponential distribution is thatA property of the exponential distribution is that the mean, the mean, , and standard deviation, , and standard deviation, , are equal., are equal.

Thus, the standard deviation, Thus, the standard deviation, , and variance, , and variance, 22, for, forthe time between arrivals at Al’s full-service pump are:the time between arrivals at Al’s full-service pump are:

= = = 3 = 3 minutesminutes

2 2 = (3) = (3)2 2 = 9 = 9

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The exponential distribution is skewed to the right.The exponential distribution is skewed to the right. The exponential distribution is skewed to the right.The exponential distribution is skewed to the right.

The skewness measure for the exponential distributionThe skewness measure for the exponential distribution is 2.is 2. The skewness measure for the exponential distributionThe skewness measure for the exponential distribution is 2.is 2.

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Relationship between the PoissonRelationship between the Poissonand Exponential Distributionsand Exponential Distributions

The Poisson distributionThe Poisson distributionprovides an appropriate descriptionprovides an appropriate description

of the number of occurrencesof the number of occurrencesper intervalper interval

The Poisson distributionThe Poisson distributionprovides an appropriate descriptionprovides an appropriate description

of the number of occurrencesof the number of occurrencesper intervalper interval

The exponential distributionThe exponential distributionprovides an appropriate descriptionprovides an appropriate description

of the length of the intervalof the length of the intervalbetween occurrencesbetween occurrences

The exponential distributionThe exponential distributionprovides an appropriate descriptionprovides an appropriate description

of the length of the intervalof the length of the intervalbetween occurrencesbetween occurrences

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End of Chapter 6End of Chapter 6

Slides Prepared by JOHN S. LOUCKS St. Edward’s University

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