SBE10_06 [Read-Only] [Compatibility Mode]

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11SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern

Chapter 6Chapter 6Continuous Probability DistributionsContinuous Probability Distributions

■■ Uniform Probability DistributionUniform Probability Distribution

■■ Normal Probability DistributionNormal Probability Distribution

■■ Exponential Probability DistributionExponential Probability Distribution

f (x)f (x)

xx

UniformUniform

xx

f f ((xx))NormalNormal

xx

f (x)f (x) ExponentialExponential


Continuous Probability DistributionsContinuous Probability Distributions

■■ A A continuous random variablecontinuous random variable can assume any value can assume any value in an interval on the real line or in a collection of in an interval on the real line or in a collection of intervals.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 assuming a The probability of the random variable assuming a value within some given interval from value within some given interval from xx11 to to xx22 is is defined to be the defined to be the area under the grapharea under the graph of the of the probability density functionprobability density function betweenbetween xx11 andand xx22..

f (x)f (x)

xx

UniformUniform

xx11xx11 xx22xx22

xx

f f ((xx))NormalNormal

xx11xx11 xx22xx22

xx11xx11 xx22xx22

ExponentialExponential

xx

f (x)f (x)

xx11xx11 xx22xx22


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

■■ A random variable is A random variable is uniformly distributeduniformly distributedwhenever 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))22/12/12

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


■■ Expected Value of Expected Value of xx

■■ Variance of Variance of xx



■■ Example: Slater's BuffetExample: Slater's Buffet

Slater customers are chargedSlater customers are charged

for the amount of salad they take. for the amount of salad they take.

Sampling suggests that theSampling suggests that the

amount of salad taken is amount of salad taken is

uniformly distributeduniformly distributed

between 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 << 1515

= 0 elsewhere= 0 elsewhere

where:where:

xx = salad plate filling weight= salad plate filling weight



■■ 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

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

= (15 = (15 –– 5)5)22/12/12

= 8.33= 8.33


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■■ Uniform Probability DistributionUniform Probability Distribution

for Salad Plate Filling Weightfor Salad Plate Filling Weight

f(x)f(x)

xx55 1010 1515

1/101/10

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



f(x)f(x)

xx55 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 salad?will take between 12 and 15 ounces of salad?

1212


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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 continuous important distribution for describing a continuous random variable.random variable.

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


HeightsHeightsof peopleof people


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

ScientificScientificmeasurementsmeasurements

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AmountsAmounts

of rainfallof rainfall


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

TestTestscoresscores



■■ Normal Probability Density FunctionNormal Probability Density Function

2 2( ) /21( )

2

xf x e µ σ

σ π− −=

µµ = mean= mean

σσ = standard deviation= standard deviation

ππ = 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 skewnessmeasure is zero.measure is zero.


■■ CharacteristicsCharacteristics

xx


The entire family of normal probabilityThe entire family of normal probability

distributions is defined by itsdistributions is defined by its meanmean µµ and itsand itsstandard deviationstandard deviation σσ ..



Standard Deviation Standard Deviation σσ

Mean Mean µµxx

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



xx




--1010 00 2020

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

xx

10




σσσσσσσσ = 15= 15

σ σ σ σ σ σ σ σ = 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.

xx


Probabilities for the normal random variable areProbabilities for the normal random variable aregiven by given by areas under the curveareas under the curve. The total area. The total areaunder 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.

68.26%68.26%

+/+/-- 1 standard deviation1 standard deviation



95.44%95.44%

+/+/-- 2 standard deviations2 standard deviations



99.72%99.72%

+/+/-- 3 standard deviations3 standard deviations




xxµµ –– 33σσ µµ –– 11σσ

µµ –– 22σσµµ + 1+ 1σσ

µµ + 2+ 2σσµµ + 3+ 3σσµµ

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 distributionwith a mean of 0 and a standard deviation of 1 iswith a mean of 0 and a standard deviation of 1 issaid to have a said to have a standard normal probabilitystandard normal probabilitydistributiondistribution..


σ σ = 1= 1

00zz

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


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■■ Converting to the Standard Normal DistributionConverting to the Standard Normal Distribution


zx= − µ

σ

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



■■ Standard Normal Density FunctionStandard Normal Density Function

π−=

2 /21( )

2

zf x e

z z = (= (xx –– µµ)/)/σσππ = 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 includingPep Zone sells auto parts and supplies including

a popular multia popular multi--grade motor oil. When thegrade 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.Pep

Zone

5w-20Motor Oil


The store manager is concerned that sales are beingThe store manager is concerned that sales are 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 leadreplenishment lead--time is normallytime 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).


PepZone

5w-20Motor Oil

■■ Example: Pep ZoneExample: Pep Zone

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zz = (= (xx -- µµ)/)/σσ= (20 = (20 -- 15)/615)/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.

PepZone

5w-20Motor Oil

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

see next slidesee next slide



■■ 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 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224

.6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549

.7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852

.8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133

.9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389

. . . . . . . . . . .

PepZone

5w-20Motor Oil

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


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PP((z z > .83) = 1 > .83) = 1 –– PP((zz << .83) .83)

= 1= 1-- .7967.7967

= .2033= .2033


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

PepZone

5w-20Motor Oil

ProbabilityProbabilityof a stockoutof a stockout PP((xx > 20)> 20)




00 .83.83

Area = .7967Area = .7967Area = 1 Area = 1 -- .7967.7967

= .2033= .2033

zz

PepZone

5w-20Motor Oil


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■■ Standard Normal Probability DistributionStandard Normal Probability Distribution

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

PepZone

5w-20Motor Oil



■■ Solving for the Reorder PointSolving for the Reorder Point

PepZone

5w-20Motor Oil

00

Area = .9500Area = .9500

Area = .0500Area = .0500

zzzz.05.05


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PepZone

5w-20Motor Oil

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

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

. . . . . . . . . . .

1.5 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .9441

1.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .9545

1.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .9633

1.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .9706

1.9 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .9767

. . . . . . . . . . .We look up the complement We look up the complement of the tail area (1 of the tail area (1 -- .05 = .95).05 = .95)




PepZone

5w-20Motor Oil

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

xx = = µµ + + zz.05.05σσ = 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 probabilityof a stockout during leadtime at (slightly less than) .05.of a stockout during leadtime at (slightly less than) .05.


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PepZone

5w-20Motor 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.



Normal Approximation Normal Approximation of Binomial Probabilitiesof Binomial Probabilities

�� When the number of trials, When the number of trials, nn, becomes large,, becomes large,evaluating the binomial probability function byevaluating the binomial probability function byhand or with a calculator is difficulthand or with a calculator is difficult

�� The normal probability distribution provides anThe normal probability distribution provides aneasyeasy--toto--use approximation of binomial probabilitiesuse approximation of binomial probabilitieswhere where nn > 20, > 20, np np >> 5, and 5, and nn(1 (1 -- pp) ) >> 5.5.

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Normal Approximation Normal Approximation of Binomial Probabilitiesof Binomial Probabilities

■■ Set Set µµ = = npnp

σ = −(1 )np p

■■ Add and subtract 0.5 (a Add and subtract 0.5 (a continuity correction factorcontinuity correction factor))

because a continuous distribution is being used tobecause a continuous distribution is being used to

approximate a discrete distribution. For example, approximate a discrete distribution. For example,

PP((xx = 10) is approximated by = 10) is approximated by PP(9.5 (9.5 << xx << 10.5).10.5).


Exponential Probability DistributionExponential Probability Distribution

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

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

Time betweenTime betweenvehicle arrivalsvehicle arrivalsat a toll boothat a toll booth

Time requiredTime requiredto completeto complete

a questionnairea questionnaire

Distance betweenDistance betweenmajor defectsmajor defectsin a highwayin a highway

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■■ Density FunctionDensity Function


where: where: µµ = mean= mean

ee = 2.71828= 2.71828

f x e x( ) /= −1µ

µfor for xx >> 0, 0, µµ > 0> 0


■■ Cumulative ProbabilitiesCumulative Probabilities


P x x e x( ) /≤ = − −0 1 o µ

where:where:

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

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■■ Example: Al’s FullExample: Al’s Full--Service PumpService Pump

The time between arrivals of carsThe time between arrivals of cars

at Al’s fullat Al’s full--service gas pump followsservice gas pump follows

an exponential probability distributionan exponential probability distribution

with 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 the

probability that the time between two successiveprobability that the time between two successive

arrivals will be 2 minutes or less.arrivals will be 2 minutes or less.


xx

f(x)f(x)

.1.1

.3.3

.4.4

.2.2

1 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8 9 10Time Between Successive Arrivals (mins.)Time Between Successive Arrivals (mins.)


PP((xx << 2) = 1 2) = 1 -- 2.718282.71828--2/32/3 = 1 = 1 -- .5134 = .4866.5134 = .4866

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

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

σσ = = µµ = 3 minutes= 3 minutes

σσ 2 2 = (3)= (3)2 2 = 9= 9



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 distributionis 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 exponential distributionThe exponential distributionprovides an appropriate descriptionprovides an appropriate description

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


End of Chapter 6End of Chapter 6

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