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Distributions

# Distributions - Standard Discrete Distributions Binomial...

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Standard Discrete Distributions Binomial Distribution Hypergeometric Distribution Poisson Distribution Geometric Distribution and their applications.

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Bernoulli Random Variable Suppose that a trial, or an experiment, whose outcome can be classified as either a success or a failure is performed. If we let X=1 when the outcome is a success and X=0 when the outcome is a failure, then the pmf of X is given by 1 ( ) (1 ) 0,1. where p, 0 p 1, is the probability that the trial is a success. x x f x p p x - = - =
A random variable X is said to be a Bernoulli random variable (after the Swiss mathematician James Bernoulli) if its probability mass function is given by 1 ( ) (1 ) 0,1. where p, 0 p 1, is the probability that the trial is a success. x x f x p p x - = - =

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Binomial Random Variable Suppose now that n independent trials, each of which results in a success with probability p and in a failure with probability 1-p , are to be performed. If X represents the number of successes that occur in the n trials, then X is said to be a Binomial random variable with parameters (n,p) . Thus a Bernoulli random variable is just a binomial random variable with parameters (1,p) .
Binomial Distribution Bernoulli Trials 1. There are only two possible outcomes for each trial. 2. The probability of a success is the same for each trial. 3. There are n trials, where n is a constant. 4. The n trials are independent.

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Binomial Distribution (cont’d) Let X be the random variable that equals the number of successes in n trials. If p and 1 – p are the probabilities of success and failure on any one trial then the probability of getting x successes and n – x failures in some specific order is p x (1- p ) n – x The number of ways in which one can select the x trials on which there is to be a success is x n
Binomial Distribution (cont’d) Thus the probability of getting x successes in n trials is given by n x p p x n p n x b x n x .... , , 2 , 1 , 0 for ) 1 ( ) , ; ( = - = - This probability distribution is called the binomial distribution because for x = 0, 1, 2, …, and n the value of the probabilities are successive terms of binomial expansion of [ p + (1 – p )] n ;

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Binomial Distribution (cont’d) for the same reason, the combinatorial quantities are referred to as binomial coefficients. The preceding equation defines a family of probability distributions with each member characterized by a given value of the parameter p and the number of trials n . x n
Example An experiment consists of four tosses of a coin. Denoting the outcomes HHTH, THTT, …., and assuming that all 16 outcomes are equally likely, find the probability distribution for the total number of head. Solution: n = 4 and p = 1/2, Let X be the total number of heads in 4 tosses where x = 0, 1, 2, 3, 4, the probability distribution for X is given by f ( x ) = P ( X = x) = b ( x :4,1/2)

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16 1 ) 4 ( , 16 4 ) 3 ( 16 6 ) 2 ( 16 4 ) 1 ( , 16 1 ) 0 ( 4 , 3 , 2 , 1 , 0 for 2 1 )! 4 ( !
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Distributions - Standard Discrete Distributions Binomial...

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