FormulaSheet

FormulaSheet - α Note that Γ α 1 = α Γ α for α>...

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Formula sheet The Normal Equations E ( X n +1 )= ° α 0 + n ± j =1 ° α j E ( X j ) Cov ( X i ,X n +1 )= n ± j =1 ° α j Cov ( X i ,X j ) Empirical Bayesian Estimation Formulas ˆ μ = ¯ X υ i = ² n i j =1 m ij ( X ij X i ) 2 n i 1 υ = ² r i =1 ( n i 1)ˆ v i ² r i =1 ( n i 1) ˆ a =( m m 1 r ± i =1 m 2 i ) 1 [ r ± i =1 m i ( ¯ X i ¯ X ) 2 ˆ υ ( r 1)] Distributions Summary X is Binomial with parameters n (positive integer) and p (0 , 1) p.f. f ( x )= ³ n x ´ p x (1 p ) n x for x =0 , 1 , 2 , 3 ,...... n E ( X )= np, V ar ( X )= np (1 p ) X is Bernoulli with parameters p (0 , 1) X is Binomial with n =1 X is Poisson with parameter λ > 0 p.f. f ( x )= λ x e λ x ! for x =0 , 1 , 2 , 3 ,...... E ( X )= λ ,V a r ( X )= λ X is Gamma with parameters α > 0 and β > 0 p.f. f ( x )= x α 1 e x/ β Γ ( α ) β α for x > 0 E ( X k )= β k Γ ( α + k ) Γ ( α ) , if k> α
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Unformatted text preview: α Note that Γ ( α + 1) = α Γ ( α ) for α > and Γ ( α + 1) = α ! for α = 0 , 1 , 2 ,.... • X is Exponential with parameters β > → X is Gamma with α = 1 • X is Pareto with parameters α > 0 and β > p.f. f ( x ) = αβ α ( x + β ) α +1 for x > c.d.f. F ( x ) = 1 − ( β x + β ) α for x ≥ E ( X ) = β α − 1 for α > 1 , V ar ( X ) = αβ 2 ( α − 1) 2 ( α − 2) for α > 2...
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