Lec11 - IOE/Stat IOE/Stat 265, Fall 2009 Lecture #11:...

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IOE/Stat 265, Fall 2009 Lecture #11: Gamma, Gamma, Weibull Weibull, and , and LogNormal LogNormal, and Beta Distributions , and Beta Distributions Ch. 4 - 4, 5 1 NOTE: Topic in Section 4.6 Probability Plots were discussed in Lecture #2 amma Distribution Properties Gamma Distribution Properties amma family presents a variety of skewed ± Gamma family – represents a variety of skewed distributions based on a shape and scale parameter. ± Shape parameter~ α ; Scale parameter ~ β 0.8 1.0 α=2; β=1/3 Scale Parameter: β < 1 compresses f(x) in x β = 1 standard gamma 0.6 f(x; ) α=1; β=1 α=2; β=1 g β > 1 stretches f(x) in x Shape Parameter: 1 f(x) strictly 0.2 0.4 α=2; β=2 α 1 f(x) strictly decreases α > 1 f(x) rises to a maximum and then ecreases 2 0.0 02468 decreases MPORTANT) NOTATION (IMPORTANT) NOTATION DeVore Textbook vs. Minitab vs. Excel ± α = r = “shape” parameter ± β = 1/ λ = “scale” parameter ± θ = “threshold” parameter 3 rigins of Gamma/Erlang Distributions Origins of Gamma/Erlang Distributions ± Exponential describes time to first random event. ± Erlang describes the time to the “r th ” event. ± If we let r be any non-negative value then we y g have Gamma. 4
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amma Function Gamma Function or 0 gamma function is defined by: ± For α > 0, gamma function is defined by: α− α 1x ) x ed x ( 46 ) Γα = 0 () x (4.6) ± Useful gamma function properties: ± For any α > 1, Γ(α) = (α−1) ∗ Γ(α−1) ± For any positive integer, n, Γ( n) = (n-1)! (1 / 2) Γ= π 5 (/) Gamma Distribution / tandard Gamma Standard Gamma amma distribution pdf ± Gamma distribution pdf ± Satisfies both conditions of a pdf. ± If α takes an integer value Æ gamma is often known as rlang distribution Erlang distribution Gamma Distribution ( β >0) 1 xe x0 fx ;, t h i β α βΓα αβ = Note: α > 0; β > 0 E(X) = μ = αβ = r / λ 0 otherwise 6 Standard Gamma Distribution ( β =1) V(X) = σ 2 = αβ 2 = r / λ 2 omputing Probabilities Using Gamma Computing Probabilities Using Gamma uppose X is a continuous rv then the cdf for the ± Suppose X is a continuous rv, then the cdf for the standard gamma rv is: x 1y ye (x; ) dy x 0 > bove equation also known as incomplete gamma function. 0 F(x; ) dy α = ± Above equation also known as incomplete gamma function. ± For non-standard Gamma, the cdf can be found as: x F(x; , ) F ; ⎛⎞ α ⎜⎟ 7 where F ; is the incomplete gamma function β ⎝⎠ ⋅α amma Applications Gamma Applications eliability Assessment Queuing Theory ± Reliability Assessment, Queuing Theory, Computer Evaluations, Biological Studies ± X usually represents the time of occurrence of usua y ep ese ts t e t e o occu e ce o an event.
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This note was uploaded on 03/17/2010 for the course IOE 265 taught by Professor Garyherrin during the Fall '09 term at University of Michigan-Dearborn.

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Lec11 - IOE/Stat IOE/Stat 265, Fall 2009 Lecture #11:...

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