Chapter4_Lecture4

OutlineGamma Distribution

Exponential DistributionOther Distributions

Exercises

Chapter 4 - Lecture 4The Gamma Distribution and its Relatives

Andreas Artemiou

Novemer 2nd, 2009

Andreas Artemiou Chapter 4 - Lecture 4 The Gamma Distribution and its Relatives



Exercises

Gamma DistributionGamma functionProbability distribution functionMoments and moment generating functionsCumulative Distribution Function

Exponential DistributionDefinitionMoments, moment generating function and cumulativedistribution function

Other Distributions

Exercises




Exercises

Gamma functionProbability distribution functionMoments and moment generating functionsCumulative Distribution Function

Gamma Function

I In this lecture we will use a lot the gamma function.I For > 0 the gamma function is defined as follows:

() =

0

x1exdx

I Properties of gamma function:I () = ( 1)( 1)I For integer n, (n) = (n 1)!I

(1

2

)=pi




Exercises


Gamma Distribution

I If X is a continuous random variable then is said to have agamma distribution if the pdf of X is:

f (x ;, ) =

1

()x1e

x

, x 00, otherwise

I If = 1 then we have the standard gamma distribution.




Exercises


Mean, Variance and mgf

I Mean: E (X ) = I Variance: var(X ) = 2

I Mgf: MX (t) =1

(1 t)




Exercises


Cumulative Distribution Function

I When X follows the standard Gamma distribution then its cdfis:

F (x ;) =

x0

y1ey

()dy , x > 0

I This is also called the incomplete gamma functionI If X (, ) then:

F (x ;, ) = P(X x) = F(x

;

)




Exercises


Example 4.27 page 193




Exercises

DefinitionMoments, moment generating function and cumulative distribution function

Exponential Distribution

I The exponential distribution is a special case of Gamma. That

is if: X Exp() X (

1,1

)I If X is a continuous random variable is said to have an

exponential distribution with parameter > 0 if the pdf of Xis:

f (x ;) =

{ex , x > 00, otherwise




Exercises


Mean, Variance mgf and cdf

I Mean: E (X ) =1

I Variance: var(X ) =1

2

I Mgf: MX (t) =1(

1 1t

)I F (x) = 1 ex , x 0




Exercises


Example 4.28 page 195




Exercises

Other useful distributions

I Chi - square distributionI t distributionI F distributionI Log - normal distributionI Beta distributionI Weibull distribution




Exercises

Exercises

I Section 4.4 page 197I Exercises 69, 70, 71, 72, 73, 74, 75, 76, 78, 80, 81


OutlineGamma DistributionGamma functionProbability distribution functionMoments and moment generating functionsCumulative Distribution Function

Exponential DistributionDefinitionMoments, moment generating function and cumulative distribution function

Other DistributionsExercises

Documents

Chapter4_Lecture4