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

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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 Gamma Distribution Properties Gamma family represents 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 β > 1 stretches f(x) in x Shape Parameter: α 1 f(x) strictly 0.2 0.4 f α=2; β=2 1 f(x) strictly decreases α > 1 f(x) rises to a maximum and then decreases 2 0.0 0 2 4 6 8 (IMPORTANT) NOTATION DeVore Textbook vs. Minitab vs. Excel α = r = “shape” parameter β = 1/ λ = “scale” parameter θ = “threshold” parameter 3 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 have Gamma. 4

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Gamma Function For > 0 gamma function is defined by: For α > 0, gamma function is defined by: α− Γ α 1 x ( ) x e dx (4 6) = 0 e dx (4.6) Useful gamma function properties: For any α > 1, Γ(α) = (α−1) ∗ Γ(α−1) F iti i t Γ( ) ( 1)! For any positive integer, n, n) = (n-1)! (1 / 2) Γ = π 5 Gamma Distribution / Standard Gamma Gamma distribution pdf Gamma distribution pdf Satisfies both conditions of a pdf. If α takes an integer value Æ gamma is often known as Erlang distribution Gamma Distribution ( β >0) 1 ( ) ( ) 1 x x e x 0 f x; , 0 th i α− β α β Γ α α β = Note: α > 0; β > 0 E(X) = μ = αβ = r / λ otherwise 6 Standard Gamma Distribution ( β =1) V(X) = σ 2 = αβ 2 = r / λ 2 Computing Probabilities Using Gamma Suppose 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 1 y y e F(x; ) dy x 0 α− α > Above equation also known as incomplete gamma function. 0 dy ( ) = Γ α Above equation also known as incomplete gamma function. For non-standard Gamma, the cdf can be found as: x F(x; , ) F ; α β = α β 7 ( ) where F ; is the incomplete gamma function α Gamma Applications Reliability Assessment Queuing Theory Reliability Assessment, Queuing Theory, Computer Evaluations, Biological Studies X usually represents the time of occurrence of an event. Example: Emails arrive to a server. After a i b f il i h il certain number of emails arrive, the emails are released to the individual users.
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Lec11 - IOE/Stat IOE/Stat 265 Fall 2009 Lecture#11 Gamma...

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