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Special Continuous Probability Distributions

Gamma DistributionBeta Distribution

Dr. Jerrell T. Stracener, SAE Fellow

Leadership in Engineering

EMIS 7370/5370 STAT 5340 : PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS

Systems Engineering ProgramDepartment of Engineering Management, Information and Systems

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Gamma Distribution

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• A family of probability density functions that yields a wide variety of skewed distributional shapes is theGamma Family.

• To define the family of gamma distributions, we first need to introduce a function that plays an important role in many branches of mathematics, i.e., the GammaFunction

The Gamma Distribution

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• Definition

For , the gamma function is defined by

• Properties of the gamma function:

1. For any [via integration by parts]

2. For any positive integer,

3.

)1()1()(,1

)!1()(, nnn

2

1

0 )(

0

1)( dxex x

Gamma Function

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Family of Gamma Distributions

• The gamma distribution defines a family of which other distributions are special cases.

• Important applications in waiting time and reliability analysis.

• Special cases of the Gamma Distribution– Exponential Distribution when α = 1– Chi-squared Distribution when

,22

and

Where is a positive integer

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A continuous random variable is said to have a gamma distribution if the probability density function of is

where the parameters and satisfy

The standard gamma distribution has

The parameter is called the scale parameter because values other than 1 either stretch or compress the probability density function.

,0)(

1 1

xforex

x

otherwise,

),;( xf

XX

.0,0

1

0

Gamma Distribution - Definition

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Standard Gamma Distribution

The standard gamma distribution has 1

The probability density function of the standard Gamma distribution is:

xexxf

1

)(

1);(

0xfor

And is 0 otherwise

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Gamma density functions

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Standard gamma density functions

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If

the probability distribution function of is

for y=x/β and x ≥ 0.

Then use table of incomplete gamma function in Appendix A.24 in textbook for quick computation of probability of gamma distribution.

X

);()(

1)()( *

0

1

yFdyeyxXPxF y

thenGX ),,( ~

Probability Distribution Function

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12

•Mean or Expected Value

•Standard Deviation

)(XE

Gamma Distribution - Properties

),( If x ~ G , then

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Suppose the reaction time of a randomly selected

individual to a certain stimulus has a standard

gamma distribution with α = 2 sec. Find the

probability that reaction time will be

(a) between 3 and 5 seconds

(b) greater than 4 seconds

Solution

Since

X

)2;3()2;5()3()5()53( ** FFFFXP

Gamma Distribution - Example

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The probability that the reaction time is more than 4 sec is

092.0

908.01)2;4(1)4(1)4( *

FXPXP

960.02

1)2;5(

801.02

1)2;3(

5

0

*

3

0

*

dyyeF

dyyeF

y

y

Gamma Distribution – Example (continued)

Where

and

159.0801.0960.0)53( xP

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Incomplete Gamma Function

Let X have a gamma distribution with parameters and .

Then for any x>0, the cdf of X is given by

Where is the incomplete gamma function.

);(),;()( *

xFxFxXP

MINTAB and other statistical packages will calculate once values of x, , and have been specified.

),;( xF

);(* x

F

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Suppose the survival time X in weeks of a randomly selected male mouse exposed to 240 rads of gamma radiation has a gamma distribution with and8 15

The expected survival time is E(X)=(8)(15) = 120 weeks

43.42)15)(8( 2 and weeks

The probability that a mouse survives between 60 and 120 weeks is

)60()120()12060( XPXPXP

496.0

051.0547.0

)8;4()8;8(

)15,8;60()15,8;120(**

FF

FF

Example

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The probability that a mouse survives at least 30 weeks is

)30(1)30(1)30( XPXPXP

999.0

001.01

)8;2(1

)15,8;30(1*

F

F

Example - continue

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Beta Distribution

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Beta Distribution - Definition

A random variable is said to have a beta distribution

with parameters, , , and if

the probability density function of is

X

, A B

X

0

,)()(

)(1

),,,;(11

otherwise,isand

BxAfor

AB

xB

AB

Ax

AB

BAxf

0,0 where

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Standard Beta Distribution

11 )1()()(

)(),;(

xxxf 10 xfor

and 0 otherwise

If X ~ B( , A, B), A =0 and B=1, then X is said to have a

standard beta distribution with probability density function

,

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Graphs of standard beta probability density function

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ABA

1

AB

Beta Distribution – Properties

If X ~ B( , A, B), , then

•Mean or expected value

•Standard deviation

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Project managers often use a method labeled PERT for

Program Evaluation and Review Technique to coordinate

the various activities making up a large project. A

standard assumption in PERT analysis is that the time

necessary to complete any particular activity once it has

been started has a beta distribution with A = the

optimistic time (if everything goes well) and B = the

pessimistic time (If everything goes badly). Suppose that

in constructing a single-family house, the time (in

days) necessary for laying the foundation has a beta

distribution with A = 2, B = 5, α = 2, and β = 3. Then

X

Beta Distribution – Example

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, so For these values of α

and β, the probability density functions of is a simple

polynomial function. The probability that it takes at most

3 days to lay the foundation is

.2.3)4.0)(3(2)( XE

X

dxxx

XP23

2 3

5

3

2

!2!1

!4

3

1)3(

.407.027

11

4

11

27

452

27

4 3

2

2 xx

4.

Beta Distribution – Example (continue)