307_outline_100608

307_outline_100608 - .Thisisawaytoextendn!.Probablynewto...

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Lecture Oct 6 CORRECTED!! Last time looked at uniform distribution and exponential distribution Today get after gamma distribution Background—the gamma function. This is a way to extend n! to all positive numbers. Probably new to you—not covered in calculus For 0, α > define ( ) ( ) 1 0 exp x x dx α α Γ = Who would come up with this? Comes up in 19 th century applied math (precomputer applied math) a lot. Learn about it. Properties: ( ) 1 1. Γ = (easy verification) ( ) ( ) 1 α α α Γ + = Γ (integrate by parts—we do this at board) Corollary: for any positive integer n, ( ) ( ) 1 ! n n Γ = Another fact: ( ) 1 2 π Γ = not that hard to see but we want to do this later in connection with other stuff. Example: Find ( ) 7 2 Γ ANS: ( ) ( ) ( ) 5 2 3 2 1 2 π Formula: ( ) ( ) 1 0 exp x x dx α α α λ λ Γ = Proof: Let . y x λ = then dy dx λ = and so ( ) ( ) ( ) 1 1 0 0 exp exp x x dx y y dy α α α α α λ λ λ Γ = = THEREFORE for fixed 0, 0 α λ > > ( ) ( ) ( ) 1 exp , 0, x f x x x α α λ λ α = > Γ defines a density—the GAMMA DENSITY with parameters ,
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