continuous probability distributions n uniform probability distribution n normal probability...

Continuous Probability DistributionsContinuous 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


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 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

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:

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

Expected Value of Expected Value of xx

Variance of Variance of xx

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

Uniform Probability DistributionUniform Probability Distribution

Example: Slater's BuffetExample: Slater's Buffet

Uniform Probability DistributionUniform Probability DistributionSlater 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 theamount of salad taken is amount of salad taken is uniformly distributeduniformly distributedbetween 5 ounces and 15 ounces.between 5 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

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

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

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.)


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?

1212

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.

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

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


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, and is , and is bell-shapedbell-shaped.. The distribution is The distribution is symmetricsymmetric, and is , and is bell-shapedbell-shaped..


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 . .

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.. 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 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



= 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

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



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



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..

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. 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 ..

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

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

NORMSINVNORMSINVNORMSINVNORMSINVNORM S INVNORM S INV

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

is used to compute the cumulativeis used to compute the cumulative probability given a probability given a zz value. value.NORMSDISTNORMSDISTNORMSDISTNORMSDISTNORM S DISTNORM S DIST

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

Formula WorksheetFormula Worksheet


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

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



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


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 suppliesPep Zone sells auto parts and supplies

including a popular multi-grade motorincluding a popular multi-grade motor

oil. When the stock of this oil drops tooil. When the stock of this oil drops to

20 gallons, a replenishment order is20 gallons, a replenishment order is

placed.placed.

PepZone5w-20Motor Oil



The store manager is concerned that The store manager is concerned that sales are being lost due to stockouts while sales are being lost due to stockouts while waiting for an order. It has been determined waiting for an order. It has been determined that demand during replenishment leadtime is that demand during replenishment leadtime is normally distributed with a mean of 15 gallons normally distributed with a mean of 15 gallons and a standard deviation of 6 gallons. and a standard deviation of 6 gallons.

The manager would like to know the The manager would like to know the probability of a stockout, probability of a stockout, PP((xx > 20). > 20).

PepZone5w-20

Motor Oil

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 between the mean and curve between the mean and zz = .83. = .83.Step 2: Find the area under the standard normalStep 2: Find the area under the standard normal curve between the mean and curve between the mean and zz = .83. = .83.

see next slidesee next slide see next slidesee next slide

Probability Table for theProbability Table for theStandard Normal DistributionStandard Normal Distribution

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

. . . . . . . . . . .

.5 .1915 .1695 .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 .1695 .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

. . . . . . . . . . .

Example: Pep ZoneExample: Pep ZonePep

Zone5w-20

Motor Oil

PP(0 (0 << zz << .83) .83)

PP((z z > .83) = .5 – > .83) = .5 – PP(0 (0 << zz << .83) .83) = 1- .2967= 1- .2967

= .2033= .2033

PP((z z > .83) = .5 – > .83) = .5 – PP(0 (0 << zz << .83) .83) = 1- .2967= 1- .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)



00 .83.83

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

= .2033= .2033

zz

PepZone5w-20

Motor Oil


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


Solving for the Reorder PointSolving for the Reorder Point

PepZone5w-20

Motor Oil

00

Area = .4500Area = .4500

Area = .0500Area = .0500

zzzz.05.05




Zone5w-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 the We look up the

areaarea(.5 - .05 = .45)(.5 - .05 = .45)



Zone5w-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.



Zone5w-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.

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:

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

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

PepZone5w-20

Motor Oil



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

PepZone5w-20

Motor Oil

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

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

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 EXPONDISTEXPONDIST function can be used to compute function can be used to compute exponential probabilities.exponential probabilities. The The EXPONDISTEXPONDIST function can be used to compute function can be used to compute exponential probabilities.exponential probabilities.

The EXPONDIST function has The EXPONDIST function has three argumentsthree arguments:: The EXPONDIST function has The EXPONDIST function has three argumentsthree arguments::

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

22ndnd 1/1/ 22ndnd 1/1/

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

probabilityprobability



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



A B

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


Exponential Probability DistributionExponential Probability DistributionThe time between arrivals of carsThe time between arrivals of cars

at 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 probability that the time between two successivesuccessivearrivals will be 2 minutes or less.arrivals will be 2 minutes or less.

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


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.)

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

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



A B

12 3 P (x < 2) =EXPONDIST(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 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

continuous probability distributions n uniform probability distribution n normal probability...

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