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# rv - Discrete Probability Distributions NOTE This document...

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Discrete Probability Distributions NOTE : This document could contain errors, please be warned. The text could possibly use other notation and format. Also note that the PMF is P { X = x } , we call it p ( x ) but others call it f ( x ). 1. Discrete Uniform Distribution Description : If a random variable X assumes the values x 1 , x 2 , . . . , x k , with equal probabilities, then X is a discrete uniform random variable. PMF : p ( x ) = 1 k , x = x 1 , x 2 , . . . , x k . Mean : E [ X ] = k i =1 x i k Variance : V [ X ] = k i =1 ( x i - E [ X ]) 2 k 2. Bernoulli Distribution Description : A trial can result in a success with probability p and a failure with proba- bility q ( q = 1 - p ). Then the random variable X , which takes on 0 if the trial is a failure and 1 if the trial is a success, is called the Bernoulli random variable with parameter p . PMF : p ( x ) = px + (1 - p )(1 - x ) , x = 0 , 1 . Mean : E [ X ] = p Variance : V [ X ] = p (1 - p ) 3. Binomial Distribution Description : A trial can result in a success with probability p and a failure with proba- bility q ( q = 1 - p ). Then the random variable X , the number of successes in n independent trials is called the binomial random variable with parameters n and p . PMF : p ( x ) = ( n x ) p x q n - x , x = 0 , 1 , 2 , . . . , n. Mean : E [ X ] = np Variance : V [ X ] = npq 1

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4. Geometric Distribution Description : A trial can result in a success with probability p and a failure with prob- ability q ( q = 1 - p ). Then the random variable
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