Lecture20_Chapter4

Lecture20_Chapter4 - Lecture 20 Chapter 4 Monday, November...

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Lecture 20 – Chapter 4 Monday, November 3 rd
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Gamma function ± In this lecture we will use a lot the gamma function. ± For the gamma function is defined as follows: ± Properties of gamma function: ² ² For integer n, ² 0 α> () 1 0 x x ed x α −− Γ= ( ) ( ) 11 αα Γ ( ) ( ) 1! nn Γ =− 1 2 π ⎛⎞ ⎜⎟ ⎝⎠
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Gamma Distribution ± If X is a continuous random variable then is said to have a gamma distribution if the pdf of X is: ± If then we have the standard gamma distribution () 1 1 ,0 ;, 0, x xe x fx otherwise α β αβ βα = Γ 1 β=
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Mean, Variance, mgf ± Mean: ± Variance: ± Moment generating function: ( ) EX αβ = ( ) 2 VX = () 1 1 X Mt t α β =
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CDF ± When X follows the standard Gamma distribution then its cdf is: ± This is also called the incomplete gamma function () 1 0 ;, 0 x y ye Fx d y x α −− = > Γ
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CDF ± If then ( ) ~, X Gamma αβ () ( ) ;, ; x Fx PX x F α βα β ⎛⎞ =≤ = ⎜⎟ ⎝⎠
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Example 4.27 page 193
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Exponential Distribution ± The exponential distribution is a special case of Gamma.
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This note was uploaded on 09/23/2009 for the course STAT 318 taught by Professor Staff during the Fall '08 term at Penn State.

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Lecture20_Chapter4 - Lecture 20 Chapter 4 Monday, November...

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