Probability and Stats in Computer Science: Formula Sheet - Cheat sheet for the Final Exam Discrete Distributions n px(1 p)nx for x = 0 1 n x(n x

Probability and Stats in Computer Science: Formula Sheet -...

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Cheat sheet for the Final Exam Discrete Distributions Binomial probability mass function P ( x ) = n ! x !( n x )! p x (1 p ) n - x for x = 0 , 1 , ..., n , Geometric probability mass function P ( x ) = (1 p ) x - 1 p for x = 1 , 2 , ... Poisson probability mass function P ( x ) = e - λ λ x x ! for x = 0 , 1 , ... Continuous Distributions Exponential density f ( x ) = λe - λx for 0 < x < Uniform density f ( x ) = 1 b a for a < x < b Gamma density f ( x ) = λ r Γ( r ) x r - 1 e - λx for 0 < x < Gamma-Poisson formula P ( X < x ) = P ( Y r ) ; P ( X > x ) = P ( Y < r ) for X Gamma( r, λ ), Y Poisson( λx ) Normal density f ( x ) = 1 σ 2 π e - ( x - μ ) 2 / 2 σ 2 for −∞ < x < Normal approximation Binomial( n, p ) Normal ( μ = np, σ = np (1 p ) ) for n 30, 0 . 05 p 0 . 95 Central Limit Theorem ( X 1 + . . . + X n ) σ n Normal(0,1) as n → ∞ Expected values and variances Distribution Bernoulli Binomial Geometric Poisson Exponential Gamma Uniform Normal ( p ) ( n, p ) ( p ) ( λ ) ( λ ) ( r, λ ) ( a, b ) ( μ, σ ) E ( X ) p np 1 p λ 1 λ r λ a + b 2 μ Var( X ) p (1 p ) np (1 p ) 1 p p 2 λ 1 λ 2 r λ 2 ( b a ) 2 12 σ 2 Stochastic processes Binomial process: p = λ ∆, number of frames n = t ; number of events in time t is X ( t ) Binomial( n, p ); interarrival time is T = Y ∆, where Y Geometric( p ) Poisson process: number of events in time t is X ( t ) Poisson( λt ); interarrival time is T Exponential( λ ) Markov chains k -step transition probability matrix P k = P k Steady state distribution is a solution of { πP = π π i = 1
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Queuing Systems Bernoulli single-server queuing system with capacity C Arrival probability p A = λ A ∆, departure probability p S = λ S ∆. Transition probabilities: p 00 = 1 p A , p 01 = p A ; p k,k - 1 = p S (1 p A ) , p k,k = p A p S + (1 p A )(1 p S ) , p k,k +1 = p A (1 p S ) for 1 k C 1; p C,C - 1 = p S (1 p A ) , p
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