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

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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 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 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
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Unformatted text preview: θ,η ) 1 2 θ e-| x-η | /θ η 2 θ 2 e ηt 1-θ 2 t 2 < θ 2 Normal distribution: X ∼ N ( μ,σ 2 ) f ( x ) = 1 σ √ 2 π e 1 2 σ 2 ( x-μ ) 2 ;-∞ < x < ∞ ,-∞ < μ < ∞ ,σ > Student t distribution: X ∼ t ( v ) f ( x ) = Γ ( v +1 2 ) √ vπ Γ ( v 2 ) 1 ( 1 + x 2 v ) ( v +1) / 2 ;-∞ < x < ∞ ,v = 1 , 2 ,... Chi-Square distribution: X ∼ χ 2 ( v ) f ( x ) = 1 2 v/ 2 Γ ( v 2 ) x v/ 2-1 e-x/ 2 ; x > ,v = 1 , 2 ,... F distribution: X ∼ F ( v 1 ,v 2 ) f ( x ) = Γ ( v 1 + v 2 2 ) Γ ( v 1 2 ) Γ ( v 2 2 ) ± v 1 v 2 ² v 1 / 2 x v 1 / 2-1 ; x > ,v 1 = 1 , 2 ,...,v 2 = 1 , 2 ,... 3...
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