test & solution

test & solution - , ) 1 2 e-| x- | / 2 2 e t 1- 2 t 2...

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Table 1: Special Discrete Distributions Notation and Parameters Discrete pdf f ( x ) Mean Variance MGF M X ( t ) Binomial X BIN ( n,p ) ( n x ) p x q n - x np npq ( pe t + q ) n 0 < p < 1 x = 0 , 1 ,...,n q = 1 - p Bernoulli X BIN (1 ,p ) p x q 1 - x p pq pe t + q 0 < p < 1 x = 0 , 1 q = 1 - p Negative Binomial X NB ( r,p ) ( x - 1 r - 1 ) p r q x - r r/p rq/p 2 ± pe t 1 - qe t ² r 0 < p < 1 x = r,r + 1 ,... r = 1 , 2 ,... Geometric X GEO ( p ) pq x - 1 1 /p q/p 2 pe t 1 - qe t 0 < p < 1 x = 1 , 2 ,... q = 1 - p Hypergeometric X HY P ( n,M,N ) ( M x )( N - M n - x ) / ( N n ) nM/N n M N ( 1 - M N ) N - n N - 1 Not tractable n = 1 , 2 ,...,N x = 0 , 1 ,...,n M = 0 , 1 ,...,N Poisson X POI ( μ ) e - μ μ x x ! μ μ e μ ( e t - 1) 0 < μ x = 0 , 1 ,... Discrete Uniform X DU ( N ) 1 /N N +1 2 N 2 - 1 12 1 N e t - e ( N +1) t 1 - e t N = 1 , 2 ,... x = 1 , 2 ,...,N 1
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Table 2: Special Continuous Distributions Notation and Parameters Continuous pdf f ( x ) Mean Variance MGF M X ( t ) Uniform X UNIF ( a,b ) 1 b - a a + b 2 ( b - a ) 2 12 e bt - e at ( b - a ) t a < b a < x < b Normal X N ( μ,σ 2 ) 1 2 πσ e - [( x - μ ) ] 2 / 2 μ σ 2 e μt + σ 2 t 2 / 2 0 < σ 2 Gamma X GAM ( θ,κ ) 1 θ κ Γ( κ ) x κ - 1 e - x/θ κθ κθ 2 ± 1 1 - θt ² κ 0 < θ 0 < x 0 < κ Exponential X EXP ( θ ) 1 θ e - x/θ θ θ 2 1 1 - θt 0 < θ 0 < x Two-Parameter Exponential X EXP ( θ,η ) 1 θ e - ( x - η ) η + θ θ 2 e ηt 1 - θt 0 < θ η < x Double Exponential X DE (
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Unformatted text preview: , ) 1 2 e-| x- | / 2 2 e t 1- 2 t 2 < 2 Normal distribution: X N ( , 2 ) f ( x ) = 1 2 e 1 2 2 ( x- ) 2 ;- < x < ,- < < , > Student t distribution: X t ( v ) f ( x ) = ( v +1 2 ) v ( v 2 ) 1 ( 1 + x 2 v ) ( v +1) / 2 ;- < x < ,v = 1 , 2 ,... Chi-Square distribution: X 2 ( v ) f ( x ) = 1 2 v/ 2 ( v 2 ) x v/ 2-1 e-x/ 2 ; x > ,v = 1 , 2 ,... F distribution: X F ( v 1 ,v 2 ) f ( x ) = ( v 1 + v 2 2 ) ( v 1 2 ) ( v 2 2 ) v 1 v 2 v 1 / 2 x v 1 / 2-1 ; x > ,v 1 = 1 , 2 ,...,v 2 = 1 , 2 ,... 3...
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This note was uploaded on 09/19/2011 for the course MATH 340 taught by Professor Staff during the Spring '08 term at S.F. State.

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test & solution - , ) 1 2 e-| x- | / 2 2 e t 1- 2 t 2...

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