f&tdist

f&tdist - Lecture 6 Gamma distribution, 2-distribution,...

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( ) ( ) ± ± ± Lecture 6 Gamma distribution, 2 -distribution, Student t -distribution, Fisher F -distribution. Gamma distribution. Let us take two parameters > 0 and ± > 0 . Gamma function ( ) is deFned by ( ) = x 1 e x dx. 0 If we divide both sides by ( ) we get 1 ± 1 = x 1 e x dx = y 1 e ±y dy 0 0 where we made a change of variables x = ±y. Therefore, if we deFne ² ± e ±x ( ) x 1 , x 0 f ( x | , ± ) = 0 , x < 0 then f ( x , ± ) will be a probability density function since it is nonnegative and it integrates | to one. Defnition. The distribution with p.d.f. f ( x , ± ) is called Gamma distribution with | parameters and ± and it is denoted as ( , ± ) . Next, let us recall some properties of gamma function ( ) . If we take > 1 then using integration by parts we can write: ( ) = x 1 e x dx = x 1 d ( e x ) 0 0 = x 1 ( e x ) 0 0 ( e x )( 1) x 2 dx = ( 1) x ( 1) 1 e x dx = ( 1) ( 1) . 0 35
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( ) ( ) ± ²³ ´ ± ( ) ( ) ± ²³ ´ Since for = 1 we have (1) = e x dx = 1 0 we can write (2) = 1 1 , (3) = 2 1 , (4) = 3 2 1 , (5) = 4 3 2 1 · · · · · · · and proceeding by induction we get that ( n ) = ( n 1)! Let us compute the k th moment of gamma distribution. We have, ± ± E X k k 1 e ±x dx = ( + k ) 1 e ±x dx = x x x 0 0 = ± ( + k ) ± + k x + k 1 e ±x dx ( ) ± + k 0 ( + k ) p.d.f. of ( + k, ± ) integrates to 1 ± ( + k ) ( + k ) ( + k 1) ( + k 1) = = = ( ) ± + k ( ) ± k ( ) ± k = ( + k 1)( + k 2) . . . ( ) = ( + k 1) ··· . ( ) ± k ± k Therefore, the mean is E X = the second moment is E X 2 ( + 1) = ± 2 and the variance ( + 1) µ 2 Var( X ) = E X 2 ( E X ) 2 = ± 2 ± = ± 2 . Below we will need the following property of Gamma distribution.
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This note was uploaded on 03/04/2009 for the course IEE 670 taught by Professor Rongpan during the Spring '09 term at ASU.

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f&amp;tdist - Lecture 6 Gamma distribution, 2-distribution,...

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