st371-15 cont rv 03 - ST371 Introduction to Proabability...

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ST371 S3 7 Introduction to Proabability and Distribution Theory Continuous Random Variables Gamma Models (c) Tom Gerig ST371-15 cont rv 03 Page 1
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he (Complete) Gamma Function The (Complete) Gamma Function he complete gamma function is defined a The complete gamma function is defined as: 1 () , 0 x xe d x f o r     0 (c) Tom Gerig ST371-15 cont rv 03 Page 2
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he (Complete) Gamma Function The (Complete) Gamma Function 1 0 () , 0 x xe d x f o r   .. e g   0 0 0 (1) (0 ) 1 xx ed x e e  (c) Tom Gerig ST371-15 cont rv 03 Page 3
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The (Complete) Gamma Function () ( 1 )( 1 )   12 (1 ) ux d u xd x   1 ) x e d x  xx dv e dx v e 0 xe  0 0 ((1 ) ) x e d x     ) 1 0( 1 ) ( 1 ) ( 1 ) x d x      (c) Tom Gerig ST371-15 cont rv 03 Page 4 0
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he (Complete) Gamma Function The (Complete) Gamma Function 1 0 () , 0 x xe d x f o r   ) ( 1 )( 1 ) Properties of the gamma function:  ) ( 1)! nn  ) (1/ 2) (c) Tom Gerig ST371-15 cont rv 03 Page 5
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The Gamma Distribution continuous with X pdf A continuous with rv X 1/ 1 (;, ) , 0 , x g xx e x    () for 0 and 0  is said to follow the gamma distribution with parameters and To denote such a we shall write: rv Gamma (c) Tom Gerig ST371-15 cont rv 03 Page 6 ~ ( , ) X
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The Gamma Distribution ote that ( ; ) has the form: x Note that (; . gx  1/ (;, ) , 0 x g x xC x e  1 where is chosen to insure that the () C will integrate to 1. pdf (c) Tom Gerig ST371-15 cont rv 03 Page 7
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The Gamma Distribution 1/ 1 (;, ) , 0 , () x gx x e x    00 for and  h t t l th h f th The parameter controls the shape of the
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This note was uploaded on 09/15/2011 for the course STATISTICS 371 taught by Professor Baldure during the Summer '11 term at N.C. State.

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st371-15 cont rv 03 - ST371 Introduction to Proabability...

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