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01. Summary of discrete and continuous distributions (review from Statistics 100A)

01. Summary of discrete and continuous distributions (review from Statistics 100A)

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University of California, Los Angeles Department of Statistics Statistics 100B Instructor: Nicolas Christou Probability Distributions - Summary Discrete Distributions Distribution Probability Mass Function Mean Variance Moment-generating Function Binomial P ( X = x ) = n x p x (1 - p ) n - x np np (1 - p ) [ pe t + (1 - p )] n x = 0 , 1 , · · · , n Geometric P ( X = x ) = (1 - p ) x - 1 p 1 p 1 - p p 2 pe t 1 - (1 - p ) e t x = 1 , 2 , · · · Negative Binomial P ( X = x ) = x - 1 r - 1 p r (1 - p ) x - r r p r (1 - p ) p 2 [ pe t 1 - (1 - p ) e t ] r x = r, r + 1 , · · · Hypergeometric P ( X = x ) = ( r x )( N - r n - x ) ( N n ) nr N n r N N - r N N - n N - 1 Fairly complicated! x = 0 , 1 , · · · , n if n r , x = 0 , 1 , · · · , r if n > r Poisson P ( X = x ) = λ x e - λ x ! λ λ exp [ λ ( e t - 1)] x = 0 , 1 , · · · Continuous Distributions Distribution Probability Density Function Mean Variance Moment-generating Function Uniform f ( x ) = 1 b - a a + b 2 ( b - a ) 2 12 e tb - e ta t ( b - a ) a x b Gamma f ( x ) = x α - 1 e - x β β α Γ( α ) , α, β > 0, x 0 αβ αβ
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Unformatted text preview: f ( x ) = λe-λx , λ > 0, x ≥ 1 λ 1 λ 2 (1-1 λ t )-1 Normal f ( x ) = 1 σ √ 2 π e-1 2 ( x-μ σ ) 2 μ σ 2 e μt + t 2 σ 2 2-∞ < x < + ∞ Remarks: Binomial: X represents the number of successes among n trials. Geometric: X represents the number of trials needed until the first success. Negative Binomial: X represents the number of trials needed until r successes occur. Hypergeometric: X represents the number of items among the n selected that comes from the r group. Poisson: X represents the number of events that occur in time, area, etc....
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