1 1 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated,...

1 1 Slide Slide© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.or duplicated, or posted to a publicly accessible website, in whole or in part.

Slides by

JohnLoucks

St. Edward’sUniversity



Chapter 6Chapter 6 Continuous Probability Distributions Continuous Probability Distributions

Uniform Probability DistributionUniform Probability Distribution

f (x)f (x)

x x

UniformUniform

xx

f f ((xx)) NormalNormal

xx

f (x)f (x) ExponentialExponential

Normal Probability DistributionNormal Probability Distribution Exponential Probability DistributionExponential Probability Distribution



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 It is not possible to talk about the probability of the random variable assuming a particular of the random variable assuming a particular value.value. Instead, we talk about the probability of the Instead, we talk about the probability of the random variable assuming a value within a random variable assuming a value within a given interval.given interval.



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




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 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:



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 charged for the Slater customers are charged for the amount ofamount of

salad they take. Sampling suggests that the salad they take. Sampling suggests that the amountamount

of salad taken is uniformly distributed of salad taken is uniformly distributed between 5between 5

ounces and 15 ounces.ounces and 15 ounces.



Uniform Probability Density FunctionUniform Probability Density Function

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




Expected Value of Expected Value 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 – 5)= (15 – 5)22/12/12

= 8.33= 8.33


Variance of Variance of xx



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

f(x)f(x)

x x

1/101/10

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


55 1010 151500



f(x)f(x)

x x

1/101/10

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

55 1010 151500

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?


1212



Area as a Measure of ProbabilityArea as a Measure of Probability

The area under the graph of The area under the graph of ff((xx) and ) and probability are identical.probability are identical.

This is valid for all continuous random This is valid for all continuous random variables.variables. The probability that The probability that xx takes on a value takes on a value between some lower value between some lower value xx11 and some higher and some higher value value xx22 can be found by computing the area can be found by computing the area under the graph of under the graph of ff((xx) over the interval from ) over the interval from xx11 to to xx22. .



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. It has been used in a wide variety of It has been used in a wide variety of

applicationsapplications

including:including:• Heights of Heights of peoplepeople• Rainfall Rainfall amountsamounts

• Test scoresTest scores• Scientific Scientific measurementsmeasurements Abraham de Moivre, a French mathematician, Abraham de Moivre, a French mathematician,

published published The Doctrine of ChancesThe Doctrine of Chances in 1733. in 1733. He derived the normal distribution.He derived the normal distribution.




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:



The distribution is The distribution is symmetricsymmetric; its skewness; its skewness measure 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 and its and its standard deviationstandard deviation . .



Standard Deviation Standard Deviation

Mean Mean xx



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





-10-10 00 2525

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

xx





= 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.

xx



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




Characteristics (basis for the empirical rule)Characteristics (basis for the empirical rule)




Characteristics (basis for the empirical rule)Characteristics (basis for the empirical rule)

xx – – 33 – – 11

– – 22 + 1+ 1

+ 2+ 2 + 3+ 3

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



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..




00zz

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





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 ..



Using Excel to ComputeUsing Excel to ComputeStandard Normal ProbabilitiesStandard Normal Probabilities

Excel has two functions for computing Excel has two functions for computing probabilities and probabilities and zz values for a values for a standardstandard normal distribution:normal distribution:

The “S” in the function names remindsThe “S” in the function names remindsus that they relate to the us that they relate to the standardstandardnormal probability distribution.normal probability distribution.



Excel Formula WorksheetExcel Formula Worksheet


A B12 3 P (z < 1.00) =NORM.S.DIST(1)4 P (0.00 < z < 1.00) =NORM.S.DIST(1)-NORM.S.DIST(0)5 P (0.00 < z < 1.25) =NORM.S.DIST(1.25)-NORM.S.DIST(0)6 P (-1.00 < z < 1.00) =NORM.S.DIST(1)-NORM.S.DIST(-1)7 P (z > 1.58) =1-NORM.S.DIST(1.58)8 P (z < -0.50) =NORM.S.DIST(-0.5)9

Probabilities: Standard Normal Distribution



Excel Value WorksheetExcel Value Worksheet


A B12 3 P (z < 1.00) 0.84134 P (0.00 < z < 1.00) 0.34135 P (0.00 < z < 1.25) 0.39446 P (-1.00 < z < 1.00) 0.68277 P (z > 1.58) 0.05718 P (z < -0.50) 0.30859

Probabilities: Standard Normal Distribution





A B

12 3 z value with .10 in upper tail =NORM.S.INV(0.9)4 z value with .025 in upper tail =NORM.S.INV(0.975)5 z value with .025 in lower tail =NORM.S.INV(0.025)6

Finding z Values, Given Probabilities





A B

12 3 z value with .10 in upper tail 1.284 z value with .025 in upper tail 1.965 z value with .025 in lower tail -1.966

Finding z Values, Given Probabilities




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 the a popular multi-grade motor oil. When the stock ofstock of

this oil drops to 20 gallons, a replenishment this oil drops to 20 gallons, a replenishment order isorder is

placed.placed.

The store manager is concerned that sales The store manager is concerned that sales areare

being lost due to stockouts while waiting for being lost due to stockouts while waiting for aa

replenishment order.replenishment order.



It has been determined that demand It has been determined that demand duringduring

replenishment lead-time is normally replenishment lead-time is normally distributeddistributed

with a mean of 15 gallons and a standard with a mean of 15 gallons and a standard deviationdeviation

of 6 gallons.of 6 gallons.


Example: Pep ZoneExample: Pep Zone

The manager would like to know the The manager would like to know the probabilityprobability

of a stockout during replenishment lead-time. of a stockout during replenishment lead-time. InIn

other words, what is the probability that other words, what is the probability that demanddemand

during lead-time will exceed 20 gallons? during lead-time will exceed 20 gallons? PP((xx > 20) = ? > 20) = ?



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 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




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

. . . . . . . . . . .

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

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




PP((z z > .83) = 1 – > .83) = 1 – PP((zz << .83) .83) = 1- .7967= 1- .7967

= .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.

ProbabilityProbability of a of a

stockoutstockoutPP((xx > > 20)20)





00 .83.83

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

= .2033= .2033

zz






If the manager of Pep Zone wants the If the manager of Pep Zone wants the probabilityprobability

of a stockout during replenishment lead-time of a stockout during replenishment lead-time to beto be

no more than .05, what should the reorder no more than .05, what should the reorder point be?point be?

------------------------------------------------------------------------------------------------------------------------------

(Hint: Given a probability, we can use the (Hint: Given a probability, we can use the standardstandard

normal table in an inverse fashion to find thenormal table in an inverse fashion to find the

corresponding corresponding zz value.) value.)



Solving for the Reorder PointSolving for the Reorder Point

00

Area = .9500Area = .9500

Area = .0500Area = .0500

zzzz.05.05




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

. . . . . . . . . . .


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.

We look upWe look upthe the

complement complement of the tail areaof the tail area(1 - .05 = .95)(1 - .05 = .95)





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

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.






1515xx

24.8724.87

Probability of Probability of aa

stockout stockout duringduring

replenishmenreplenishmentt

lead-time lead-time = .05= .05

Probability of Probability of aa


replenishmenreplenishmentt

lead-time lead-time = .05= .05

Probability of Probability of nono


replenishmentreplenishmentlead-time lead-time

= .95= .95

Probability of Probability of nono


replenishmentreplenishmentlead-time lead-time

= .95= .95




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 thatThis is a significant decrease in the chance thatPep Zone will be out of stock and unable to meet aPep Zone will be out of stock and unable to meet acustomer’s desire to make a purchase.customer’s desire to make a purchase.




Using Excel to ComputeUsing Excel to ComputeNormal ProbabilitiesNormal Probabilities

Excel has two functions for computing Excel has two functions for computing cumulative probabilities and cumulative probabilities and xx values for values for anyany normal distribution:normal distribution:

NORM.DISTNORM.DIST is used to compute the cumulative is used to compute the cumulativeprobability given an probability given an xx value. value.

NORM.INVNORM.INV is used to compute the is used to compute the xx value given value givena cumulative probability.a cumulative probability.





A B

12 3 P (x > 20) =1-NORM.DIST(20,15,6,TRUE)4 56 7 x value with .05 in upper tail =NORM.INV(0.95,15,6)8

Probabilities: Normal Distribution

Finding x Values, Given Probabilities





A B

12 3 P (x > 20) 0.20234 56 7 x value with .05 in upper tail 24.878

Probabilities: Normal Distribution

Finding x Values, Given Probabilities

Note: P(Note: P(xx >> 20) = .2023 here using Excel, while our 20) = .2023 here using Excel, while our previous manual approach using the previous manual approach using the zz table yielded table yielded .2033 due to our rounding of the .2033 due to our rounding of the zz value. value.



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.

•Time between vehicle arrivals at a toll boothTime between vehicle arrivals at a toll booth•Time required to complete a questionnaireTime required to complete a questionnaire•Distance between major defects in a highwayDistance between major defects in a highway

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

In waiting line applications, the exponential In waiting line applications, the exponential distribution is often used for service times.distribution is often used for service times.




A property of the exponential distribution is A property of the exponential distribution is that the mean and standard deviation are that the mean and standard deviation are equal.equal. The exponential distribution is skewed to the The exponential distribution is skewed to the right. Its skewness measure is 2.right. Its skewness measure is 2.



Density FunctionDensity Function


where: where: = expected or mean = expected or mean

ee = 2.71828 = 2.71828

f x e x( ) / 1

f x e x( ) / 1

for for xx >> 0 0



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



Using Excel to ComputeUsing Excel to ComputeExponential ProbabilitiesExponential Probabilities

The The EXPON.DISTEXPON.DIST function can be used to compute function can be used to compute exponential probabilities.exponential probabilities.

The EXPON.DIST function has The EXPON.DIST function has three argumentsthree arguments::

11stst The value of the random variable The value of the random variable xx

22ndnd 1/1/

33rdrd “TRUE” or “FALSE” “TRUE” or “FALSE”

the inverse of the the inverse of the meanmeannumber of number of occurrencesoccurrences in an intervalin an intervalWe will always enterWe will always enter

““TRUE” because we’re TRUE” because we’re seeking a cumulative seeking a cumulative

probability.probability.





A B

12 3 P (x < 18) =EXPON.DIST(18,1/15,TRUE)4 P (6 < x < 18) =EXPON.DIST(18,1/15,TRUE)-EXPON.DIST(6,1/15,TRUE)5 P (x > 8) =1-EXPON.DIST(8,1/15,TRUE)6

Probabilities: Exponential Distribution





A B

12 3 P (x < 18) 0.69884 P (6 < x < 18) 0.36915 P (x > 8) 0.58666





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

The time between arrivals of cars at Al’s The time between arrivals of cars at Al’s full-full-

service gas pump follows an exponential service gas pump follows an exponential probabilityprobability

distribution with a mean time between arrivals distribution with a mean time between arrivals of 3of 3

minutes. Al would like to know the probability minutes. Al would like to know the probability thatthat

the time between two successive arrivals will the time between two successive arrivals will be 2be 2

minutes or less.minutes or less.



xx

f(x)f(x)

.1.1

.3.3

.4.4

.2.2

0 1 2 3 4 5 6 7 8 9 10 0 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

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





A B

12 3 P (x < 2) =EXPON.DIST(2,1/3,TRUE)4






A B

12 3 P (x < 2) 0.48664




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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