ProbabilityDistributions

# ProbabilityDistributions - Stat 333 Basic Probability...

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Stat 333 Basic Probability Distributions Discrete Distributions distribution probability mass function range E( X ) Var( X ) indicator ( p ) f (0) = 1 - p , f (1) = p k = 0 , 1 p p (1 - p ) binomial( n, p ) f ( k ) = n k p k (1 - p ) n - k k = 0 , 1 , . . . , n np np (1 - p ) geometric ( p ) f ( k ) = p (1 - p ) k - 1 k = 1 , 2 , 3 , . . . 1 p 1 - p p 2 neg. binomial( r, p ) f ( k ) = k - 1 r - 1 p r (1 - p ) k - r k = r, r +1 , . . . r p r (1 - p ) p 2 Poisson( λ ) f ( k ) = λ k k ! e - λ k = 0 , 1 , 2 , . . . λ λ we will always define the geometric as X = number of trials up to (and including) the first Success. Similarly we will always define the negative binomial as X r = number of trials up to (and including) the first r Successes. The indicator, binomial, and Poisson random variables are examples of counting variables . They count the number of times a certain event E occurs in a fixed number of trials (indicator and binomial) or in a fixed time period (Poisson). The geometric and negative binomial are examples of waiting time variables . They count the number of trials (the waiting time) required to obtain a
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