Probablity normal

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© 2003 South-Western/Thomson Learning™© 2003 South-Western/Thomson Learning™

Chapter 6Chapter 6 Continuous Probability Distributions Continuous Probability Distributions

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

xx

ff((xx))

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

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

The probability of the random variable assuming The probability of the random variable assuming a value within some given interval from a 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 between between x x11 andand x x22..

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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.

Uniform Probability Density FunctionUniform Probability Density Function

ff((xx) = 1/() = 1/(bb - - aa) for ) for aa << xx << bb

= 0 = 0 elsewhere elsewhere

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

bb = largest value the variable can = largest value the variable can assumeassume

Uniform Probability DistributionUniform Probability Distribution

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Expected Value of Expected Value of xx

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

Variance of Variance of xx

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

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

bb = largest value the variable can = largest value the variable can assumeassume

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Example: Slater's BuffetExample: Slater's Buffet


Slater customers are charged for the Slater customers are charged for the amount of salad they take. Sampling suggests amount of salad they take. Sampling suggests that the amount of salad taken is uniformly that the amount of salad taken is uniformly distributed between 5 ounces and 15 ounces.distributed between 5 ounces and 15 ounces.

The probability density function isThe probability density function is

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

= 0 = 0 elsewhere elsewhere

where:where:

xx = salad plate filling weight = salad plate filling weight

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What is the probability that a customer What is the probability that a customer will will take between 12 and 15 ounces of take between 12 and 15 ounces of salad?salad?

f(x)f(x)

x x55 1010 15151212

1/101/10

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

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

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Expected Value of Expected Value of xx

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

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

= 10= 10 Variance of Variance of xx

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

= (15 – 5)= (15 – 5)22/12/12

= 8.33= 8.33

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Normal Probability DistributionNormal Probability Distribution

Graph of the Normal Probability Density Graph of the Normal Probability Density FunctionFunction

xx

ff((xx))

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Characteristics of the Normal Probability Characteristics of the Normal Probability DistributionDistribution

• The shape of the normal curve is often The shape of the normal curve is often illustrated as a illustrated as a bell-shaped curvebell-shaped curve. .

• Two parametersTwo parameters, , (mean) and (mean) and (standard (standard deviation), determine the location and shape deviation), determine the location and shape of the distribution.of the distribution.

• The The highest pointhighest point on the normal curve is at on the normal curve is at the mean, which is also the median and the mean, which is also the median and mode.mode.

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

… continued

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Characteristics of the Normal Probability Characteristics of the Normal Probability DistributionDistribution

• The normal curve is The normal curve is symmetricsymmetric..

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

• The total area under the curve is 1 (.5 to the The total area under the curve is 1 (.5 to the left of the mean and .5 to the right).left of the mean and .5 to the right).

• Probabilities for the normal random variable Probabilities for the normal random variable are given by are given by areas under the curveareas under the curve..

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Percent (%) of Values in Some Commonly Used Percent (%) of Values in Some Commonly Used IntervalsIntervals

• 68.26%68.26% of values of a normal random of values of a normal random variable are within variable are within +/- 1+/- 1 standard standard deviationdeviation of its mean. of its mean.

• 95.44%95.44% of values of a normal random of values of a normal random variable are within variable are within +/- 2+/- 2 standard standard deviationsdeviations of its mean. of its mean.

• 99.72%99.72% of values of a normal random of values of a normal random variable are within variable are within +/- 3+/- 3 standard standard deviationsdeviations of its mean. of its mean.

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Normal Probability Density FunctionNormal Probability Density Function

where:where:

= mean= mean

= standard deviation= standard deviation

= 3.14159= 3.14159

ee = 2.71828 = 2.71828

f x e x( ) ( ) / 12

2 2 2

f x e x( ) ( ) / 1

2

2 2 2

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Standard Normal Probability DistributionStandard Normal Probability Distribution

A random variable that has a normal A random variable that has a normal distribution with a mean of zero and a distribution with a mean of zero and a standard deviation of one is said to have a standard deviation of one is said to have a standard normal probability distributionstandard normal probability distribution..

The letter The letter z z is commonly used to designate is commonly used to designate this normal random variable.this normal random variable.

Converting to the Standard Normal Converting to the Standard Normal DistributionDistribution

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

zx

zx

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Using Excel to ComputeUsing Excel to Compute Standard Normal Probabilities Standard 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 normal distribution:distribution:

• NORMSDISTNORMSDIST is used to compute the is used to compute the cumulative probability given a cumulative probability given a zz value. value.

• NORMSINVNORMSINV is used to compute the is used to compute the zz value value given a cumulative probability.given a cumulative probability.

(The letter S in the above function names (The letter S in the above function names reminds usreminds us

that they relate to the standard normal that they relate to the standard normal probabilityprobability

distribution.)distribution.)

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Formula WorksheetFormula Worksheet

Using Excel to ComputeUsing Excel to ComputeStandard Normal ProbabilitiesStandard Normal Probabilities

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

Probabilities: Standard Normal Distribution

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Value WorksheetValue 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

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A B

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

Finding z Values, Given Probabilities

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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


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Example: Pep ZoneExample: Pep Zone


Pep Zone sells auto parts and supplies including aPep Zone sells auto parts and supplies including a

popular multi-grade motor oil. When the stock of thispopular multi-grade motor oil. When the stock of this

oil drops to 20 gallons, a replenishment order is placed.oil drops to 20 gallons, a replenishment order is placed.

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

lost due to stockouts while waiting for an order. It haslost due to stockouts while waiting for an order. It has

been determined that leadtime demand is normallybeen determined that leadtime demand is normally

distributed with a mean of 15 gallons and a standarddistributed with a mean of 15 gallons and a standard

deviation of 6 gallons. deviation of 6 gallons.

The manager would like to know the probability of aThe manager would like to know the probability of a

stockout, P(stockout, P(xx > 20). > 20).

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The Standard Normal table shows an area The Standard Normal table shows an area of .2967 for the region between the of .2967 for the region between the zz = 0 and = 0 and z z = .83 lines below. The shaded tail area is .5 = .83 lines below. The shaded tail area is .5 - .2967 = .2033. The probability of a stock-- .2967 = .2033. The probability of a stock-

out is .2033. out is .2033.

zz = ( = (xx - - )/)/

= (20 - 15)/6= (20 - 15)/6

= .83= .83


00 .83.83

Area = .2967Area = .2967

Area = .5Area = .5

Area = .5 - .2967Area = .5 - .2967 = .2033= .2033

zz

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Using the Standard Normal Probability TableUsing the Standard Normal Probability Table


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

.0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .0319 .0359

.1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .0714 .0753

.2 .0793 .0832 .0871 .0910 .0948 .0987 .1026 .1064 .1103 .1141

.3 .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .1480 .1517

.4 .1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .1879

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

.6 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2518 .2549

.7 .2580 .2612 .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

.0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .0319 .0359

.1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .0714 .0753

.2 .0793 .0832 .0871 .0910 .0948 .0987 .1026 .1064 .1103 .1141

.3 .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .1480 .1517

.4 .1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .1879

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

.6 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2518 .2549

.7 .2580 .2612 .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

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Standard Normal Probability DistributionStandard Normal Probability DistributionIf 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?

Let Let zz.05.05 represent the represent the zz value cutting the .05 tail value cutting the .05 tail area. area.


Area = .05Area = .05

Area = .5 Area = .5 Area = .45 Area = .45

00 zz.05.05

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Using the Standard Normal Probability TableUsing the Standard Normal Probability Table

We now look-up the .4500 area in the We now look-up the .4500 area in the Standard Normal Probability table to find the Standard Normal Probability table to find the corresponding corresponding zz.05.05 value. value.

zz.05.05 = 1.645 is a reasonable estimate. = 1.645 is a reasonable estimate.

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

. . . . . . . . . . .


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The corresponding value of The corresponding value of xx is given by is given by

xx = = + + zz.05.05

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

= 24.87= 24.87

A reorder point of 24.87 gallons will A reorder point of 24.87 gallons will place the probability of a stockout during place the probability of a stockout during leadtime at .05. leadtime at .05. Perhaps Pep Zone should set Perhaps Pep Zone should set the reorder point at 25 gallons to keep the the reorder point at 25 gallons to keep the probability under .05.probability under .05.


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Using Excel to Compute Normal Using Excel to Compute Normal ProbabilitiesProbabilities

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:

• NORMDISTNORMDIST is used to compute the is used to compute the cumulative probability given an cumulative probability given an xx value. value.

• NORMINVNORMINV is used to compute the is used to compute the xx value value given a cumulative probability.given a cumulative probability.

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Formula Worksheet for Pep Zone ExampleFormula Worksheet for Pep Zone Example


A B

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

Probabilities: Normal Distribution

Finding x Values, Given Probabilities

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Value Worksheet for Pep Zone ExampleValue Worksheet for Pep Zone Example


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.

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

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Exponential Probability DistributionExponential Probability Distribution

Exponential Probability Density FunctionExponential Probability Density Function

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

where: where: = mean = mean

ee = 2.71828 = 2.71828

f x e x( ) / 1

f x e x( ) / 1

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Cumulative Exponential Distribution FunctionCumulative Exponential Distribution Function

where:where:

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

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

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Excel’s Excel’s EXPONDISTEXPONDIST function can be used to function can be used to compute exponential probabilities.compute exponential probabilities.

The function has The function has three argumentsthree arguments::

• FirstFirst – the value of the random variable – the value of the random variable xx

• SecondSecond – 1/ – 1/ (the inverse of the mean (the inverse of the mean number of occurrences in an interval)number of occurrences in an interval)

• ThirdThird – “TRUE” or “FALSE” (we will always – “TRUE” or “FALSE” (we will always enter “TRUE” because we’re seeking a enter “TRUE” because we’re seeking a cumulative probability)cumulative probability)

Using Excel to Compute Exponential Using Excel to Compute Exponential ProbabilitiesProbabilities

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A B

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

Probabilities: Exponential Distribution

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A B

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


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The time between arrivals of cars at Al’s The time between arrivals of cars at Al’s Carwash follows an exponential probability Carwash follows an exponential probability distribution with a mean time between arrivals distribution with a mean time between arrivals of 3 minutes. Al would like to know the of 3 minutes. Al would like to know the probability that the time between two probability that the time between two successive arrivals will be 2 minutes or less.successive arrivals will be 2 minutes or less.

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

Example: Al’s CarwashExample: Al’s Carwash

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Example: Al’s CarwashExample: Al’s Carwash

Graph of the Probability Density FunctionGraph of the Probability Density Function

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 10

P(x < 2) = area = .4866P(x < 2) = area = .4866

Time Between Successive Arrivals (mins.)Time Between Successive Arrivals (mins.)

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Formula Worksheet for Al’s Carwash ExampleFormula Worksheet for Al’s Carwash Example


A B

12 3 P (x < 2) =EXPONDIST(2,1/3,TRUE)4


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Value Worksheet for Al’s Carwash ExampleValue Worksheet for Al’s Carwash Example


A B

12 3 P (x < 2) 0.48664


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Relationship between the PoissonRelationship between the Poissonand Exponential Distributionsand Exponential Distributions

(If) the Poisson distribution(If) the Poisson distributionprovides an appropriate descriptionprovides an appropriate description

of the number of occurrencesof the number of occurrencesper intervalper interval

(If) the Poisson distribution(If) the Poisson distributionprovides an appropriate descriptionprovides an appropriate description

of the number of occurrencesof the number of occurrencesper intervalper interval

(If) the exponential distribution(If) the exponential distributionprovides an appropriate descriptionprovides an appropriate description

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

(If) the exponential distribution(If) the exponential distributionprovides an appropriate descriptionprovides an appropriate description

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

Probablity normal

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