Chapter 4.4-4.6 - Statistics 511: Statistical Methods Dr....

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Unformatted text preview: Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Lecture 10: Other Continuous Distributions and Probability Plots Devore: Section 4.4-4.6 March, 2011 Page 1 Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Gamma Distribution Gamma function is a natural extension of the factorial For any > , ( ) = Z x - 1 e- x dx Properties: 1. If > 1 , ( ) = ( - 1)( - 1) 2. ( n ) = ( n- 1)! for any n Z + 3. ( 1 2 ) = March, 2011 Page 2 Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 The natural definition of a density based on the gamma function is f ( x ; ) = x - 1 e- x ( ) if x otherwise A gamma density with parameters > , > is f ( x ; , ) = 1 ( ) x - 1 e- x/ if x otherwise is a shape parameter , is a scale parameter The case = 1 is called the standard gamma distribution March, 2011 Page 3 Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Gamma pdf: graphical illustration March, 2011 Page 4 Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Gamma Distribution Parameters The mean and variance of a random variable X having the gamma distribution f ( x ; , ) are E ( X ) = and V ( X ) = 2 Let X have a gamma distribution with parameters and Then P ( X x ) = F ( x ; , ) = F ( x/ ; ) In the above, F ( x ; ) = Z x y - 1 e- y ( ) dy is an incomplete gamma function . It is defined for any x > . March, 2011 Page 5 Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Exponential Distribution as a Special Case of Gamma Distribution Assume that = 1 and = 1 . Then, f ( x ; ) = e- x if x otherwise Its mean and variance are E ( X ) = 1 and V ( X ) = 1 2 Note that = in this case March, 2011 Page 6 Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Exponential pdf: graphical illustration March, 2011 Page 7 Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Exponential cdf Exponential cdf can be easily obtained by integrating pdf, unlike the cdf of the general Gamma distribution The result is F ( x ; ) = 1- e- x if x otherwise March, 2011 Page 8 Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Example I Suppose the response time X at an on-line computer terminal has an exponential distribution with expected response time 5 sec E ( X ) = 1 = 5 and = 0 . 2 The probability that the response time is between 5 sec and 10 sec is P (5 X 10) = F (10;0 . 2)- F (5;0 . 2) = 0 . 233 March, 2011 Page 9 Statistics 511: Statistical Methods...
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This note was uploaded on 03/30/2011 for the course STAT 511 taught by Professor Bud during the Spring '08 term at Purdue University-West Lafayette.

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Chapter 4.4-4.6 - Statistics 511: Statistical Methods Dr....

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