chap5S2

# chap5S2 - Chapter5.4 Tostart, X B x=1 success x=0 failure X...

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Chapter 5.4 Binomial Distribution A special application of a discrete probability distribution is the binomial distribution. To start, introduce the random variable that takes two outcomes: B X x = 1 “success” x = 0 “failure” The probability distribution function of is: B X for 0 < p < 1 (the probability of success) p ) X ( P B = = 1 p 1 ) X ( P B = = 0 This is known as the Bernoulli distribution. The mean and variance are calculated as: p p p ) x ( P x ) X ( E x B X B = + = = = μ ) (1 0 1 ) p 1 ( p p p p p p p ) x ( P x ) X ( E ) X ( Var 2 2 2 x 2 2 X 2 B B B = = + = = μ = ) (1 (0)(0) (1)(1) Chapter 5 17

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Now consider that a random experiment with the outcome of success or failure is repeated n times. Each trial produces success or failure with probabilities p and (1 p ) respectively. Assume independence so that the result of one trial does not influence the result of any other trial. Let the random variable X be the number of successes in n trials. The probability distribution function of X is defined as: ) ( P ) x ( P trials t independen n in successes x = for x = 0, 1, 2, . . . , n This is known as the binomial distribution . Chapter 5 18