FormulaSheet

FormulaSheet - Note that ( + 1) = ( ) for > and...

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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: Note that ( + 1) = ( ) for > and ( + 1) = ! for = 0 , 1 , 2 ,.... X is Exponential with parameters > X is Gamma with = 1 X is Pareto with parameters > 0 and > p.f. f ( x ) = ( x + ) +1 for x > c.d.f. F ( x ) = 1 ( x + ) for x E ( X ) = 1 for > 1 , V ar ( X ) = 2 ( 1) 2 ( 2) for > 2...
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