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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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•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)