Distributions - Standard Discrete Distributions Binomial...

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Unformatted text preview: Standard Discrete Distributions Binomial Distribution Hypergeometric Distribution Poisson Distribution Geometric Distribution and their applications. 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- =- = 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. Binomial Distribution (contd) 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 (contd) 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 , 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 ; Binomial Distribution (contd) 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....
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This note was uploaded on 02/23/2011 for the course MATH 112 taught by Professor Ritadubey during the Spring '11 term at Amity University.

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Distributions - Standard Discrete Distributions Binomial...

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