06 Discrete probability distributions

06 Discrete probability distributions - University of...

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Unformatted text preview: University of California, Los Angeles Department of Statistics Statistics 100A Instructor: Nicolas Christou Some special discrete probability distributions • Bernoulli random variable: It is a variable that has 2 possible outcomes: “success”, or “fail- ure”. Success occurs with probability p and failure with proba- bility 1- p . 1 • Binomial probability distribution: Suppose that n independent Bernoulli trials each one having probability of success p are to be performed. Let X be the number of successes among the n trials. We say that X follows the binomial probability distribution with parameters n,p . Probability mass function of X : P ( X = x ) = n x p x (1- p ) n- x , x = 0 , 1 , 2 , 3 , ··· ,n or P ( X = x ) = nCx p x (1- p ) n- x , x = 0 , 1 , 2 , 3 , ··· ,n where nCx = n x = n ! ( n- x )! x ! Expected value of X : E ( X ) = np Variance of X : σ 2 = np (1- p ) Standard deviation of X : σ = r np (1- p 2 • Geometric probability distribution: Suppose that repeated independent Bernoulli trials each one hav- ing probability of success p are to be performed. Let X be the number of trials needed until the first success occurs. We say that X follows the geometric probability distribution with pa- rameter p . Probability mass function of X : P ( X = x ) = (1- p ) x- 1 p, x = 1 , 2 , 3 , ··· Expected value of X : E ( X ) = 1 p Variance of X : σ 2 = 1- p p 2 Standard deviation of X : σ = s 1- p p 2 3 • More on geometric probability distribution ··· Repeated Bernoulli trials are performed until the first success occurs. Find the probability that – the first success occurs after the k th trial – the first success occurs on or after the k th trial – the first success occurs before the k th trial – the first success occurs on or before the k th trial 4 • Negative binomial probability distribution: Suppose that repeated Bernoulli trials are performed until r suc- cesses occur. The number of trials required X , follows the so called negative binomial probability distribution. Probability mass function of X is: P ( X = x ) = x- 1 r- 1 p r- 1 (1- p ) x- r p, or P ( X = x ) = x- 1 r- 1 p r (1- p ) x- r x = r,r + 1 ,r + 2 , ··· 5 • Hypergeometric probability distribution: Select without replacement n from N available items (of which r are labeled as “hot items”, and N- r are labeled as “cold items”). Let X be the number of hot items among the n . Probability mass function of X : P ( X = x ) = r x N- r n- x N n 6 California Super Lotto Plus: 7 • Poisson probability distribution: The Poisson probability mass function with parameter λ > (where λ is the average number of events occur per time, area, volume, etc.) is: P ( X = x ) = λ x e- λ x !...
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This note was uploaded on 01/14/2011 for the course STATS 100A taught by Professor Wu during the Fall '07 term at UCLA.

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06 Discrete probability distributions - University of...

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