lec8 - Lecture 8 8.1 Gamma distribution. 0 Let us take two...

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8.1 Gamma distribution. Let us take two parameters α > 0 and β > 0 . Gamma function Γ( α ) is deFned by Γ( α ) = Z 0 x α - 1 e - x dx. If we divide both sides by Γ( α ) we get 1 = Z 0 1 Γ( α ) x α - 1 e - x dx = Z 0 β α Γ( α ) y α - 1 e - βy dy where we made a change of variables x = βy. Therefore, if we deFne f ( x | α, β ) = ± β α Γ( α ) x α - 1 e - βx , x 0 0 , x < 0 then f ( x | α, β ) will be a probability density function since it is nonnegative and it integrates to one. Defnition. The distribution with p.d.f. f ( x | α, β ) is called Gamma distribution with parameters α and β and it is denoted as Γ( α, β ) . Next, let us recall some properties of gamma function Γ( α ) . If we take α > 1 then using integration by parts we can write: Γ( α ) = Z 0 x α - 1 e - x dx = Z 0 x α - 1 d ( - e - x ) = x α - 1 ( - e - x ) ² ² ² 0 - Z 0 ( - e - x )( α - 1) x α - 2 dx = ( α - 1) Z 0 x ( α - 1) - 1 e - x dx = ( α - 1)Γ( α - 1) . 32
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This note was uploaded on 10/11/2009 for the course STATISTICS 18.443 taught by Professor Dmitrypanchenko during the Spring '09 term at MIT.

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lec8 - Lecture 8 8.1 Gamma distribution. 0 Let us take two...

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