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3044formulas

# 3044formulas - ISyE 3044 Important Formulas Distributions...

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ISyE 3044 — Important Formulas Distributions Parameters Density/p.m.f. c.d.f. Mean Variance Binomial n 1, 0 < p < 1 ( n k ) p k (1 - p ) n - k , k = 0 , . . . , n np np (1 - p ) Geometric 0 < p < 1 (1 - p ) k - 1 p , k = 1 , 2 , . . . Pr { X k } = 1 - (1 - p ) k 1 /p (1 - p ) /p 2 Negative Binomial n 1, 0 < p < 1 ( k - 1 n - 1 ) p n (1 - p ) k - n , k = n, n + 1 , . . . n/p n (1 - p ) /p 2 Poisson λ > 0 e - λ λ k /k !, k = 0 , 1 , . . . λ λ Uniform -∞ < a < b < 1 / ( b - a ), a x b ( x - a ) / ( b - a ) ( a + b ) / 2 ( b - a ) 2 / 12 Normal -∞ < μ < , σ > 0 1 σ 2 π e - ( x - μ ) 2 2 σ 2 , -∞ < x < Φ(( x - μ ) ) μ σ 2 Exponential λ > 0 λe - λx , x > 0 1 - e - λx 1 1 2 Gamma λ > 0, α > 0 λ ( λx ) a - 1 e - λx / Γ( α ) α/λ α/λ 2 Erlang Gamma with 1 - k - 1 i =0 e - λx ( λx ) i /i ! integer α = k Weibull λ > 0, α > 0 αλ ( λx ) α - 1 exp[ - ( λx ) α ] 1 - exp[ - ( λx ) α ], x > 0 Γ(1 + 1 ) Central Limit Theorem Pr braceleftbigg n ( ¯ X n - μ ) σ z bracerightbigg Φ( z ) as n → ∞ . Poisson Process The times between events are i.i.d. exponential with parameter λ ; The number of events in an interval [ s, s + t ] have the Poisson distribution with parameter λt ; The numbers of events in nonoverlapping intervals are independent random variables.

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3044formulas - ISyE 3044 Important Formulas Distributions...

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