Lect7_L6_L7_handout2

Lect7_L6_L7_handout2 - lecture 7 Discrete Random Variables...

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Unformatted text preview: lecture 7, Discrete Random Variables II The Binomial Distribution The Bernoulli Distribution The Binomial Distribution Examples lecture 7, Discrete Random Variables II Outline • Random Variables • Discrete Probability Distributions • Expected Value and Variance of Discrete Random Variables The Binomial Distribution • The Poisson Distribution lecture 7, Discrete Random Variables II The Binomial Distribution The Bernoulli Distribution The Binomial Distribution Examples lecture 7, Discrete Random Variables II The Binomial Distribution The Bernoulli Distribution The Bernoulli Distribution A Bernoulli trial is an experiment with two outcomes: “success” and “failure” Let p be the probability of success — The probability of failure, q , is 1- p • X = 1 if succeeds, = 0 otherwise • X is a Bernoulli ( p ) random variable lecture 7, Discrete Random Variables II The Binomial Distribution The Bernoulli Distribution The Binomial Distribution Examples lecture 7, Discrete Random Variables II The Binomial Distribution The Bernoulli Distribution Examples of Bernoulli rvs • Will the coin land head? p = 1 / 2 if the coin is fair • Will the newborn child be a girl? • Will a potential customer buy a product? • Will a voter vote for a specific candidate? lecture 7, Discrete Random Variables II The Binomial Distribution The Bernoulli Distribution The Binomial Distribution Examples lecture 7, Discrete Random Variables II The Binomial Distribution The Binomial Distribution The Binomial Distribution The binomial experiment: • Experiment consists of n identical trials • Each trial results in either “success” or “failure” • Probability of success, p , is constant from trial to trial — The probability of failure, q , is 1- p • Trials are independent If X is the total number of successes in the n trials, then X is a binomial ( n , p ) random variable Let X i =1 if the ith trial succeeds, and = 0 otherwise. Then X i is a--- Bernoulli(p) random variable, and X = X 1 + . . . + X n . lecture 7, Discrete Random Variables II The Binomial Distribution The Bernoulli Distribution The Binomial Distribution Examples lecture 7, Discrete Random Variables II The Binomial Distribution Examples Examples of Binomial rvs Number of 1’s in 10 rolls of a die is a--- Binomial ( 10 , 1 / 6 ) random variable. Number of correct guesses on a multiple-choice test (with 25 questions, 4 choices and one correct answer each) is a--- Binomial ( 25 , 1 / 4 ) random variable. Binomial or not: A couple decides to have children until they have a girl. The number of boys the couple will have is not a binomial random variable. Number of games a team will win in the next season is not a binomial random variable. lecture 7, Discrete Random Variables II The Binomial Distribution The Bernoulli Distribution The Binomial Distribution Examples lecture 7, Discrete Random Variables II The Binomial Distribution Examples Example: Marbles A box contains 1 red marble and 9 green ones. FiveA box contains 1 red marble and 9 green ones....
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Lect7_L6_L7_handout2 - lecture 7 Discrete Random Variables...

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