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# Lecture9 - lecture 9 Discrete Random Variables II 1/25 The...

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Unformatted text preview: lecture 9, Discrete Random Variables II 1/25 The Bernoulli Distribution The Binomial Distribution Examples lecture 9, Discrete Random Variables II Outline 1 The Bernoulli Distribution 2 The Binomial Distribution Examples lecture 9, Discrete Random Variables II 2/25 The Bernoulli Distribution The Binomial Distribution Examples lecture 9, Discrete Random Variables II 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 • put X = ( 1 , if succeeds, , otherwise • X is called a Bernoulli ( p ) random variable lecture 9, Discrete Random Variables II 3/25 The Bernoulli Distribution The Binomial Distribution Examples lecture 9, Discrete Random Variables II The Bernoulli Distribution Examples of Bernoulli Trials 1 Will the coin land heads? 2 Will the newborn child be a girl? 3 Will a potential customer buy a product? 4 Will a voter vote for a specific candidate? lecture 9, Discrete Random Variables II 4/25 The Bernoulli Distribution The Binomial Distribution Examples lecture 9, Discrete Random Variables II The Bernoulli Distribution Mean and Variance of Bernoulli Random Variables • Let X be a Bernoulli ( p ) random variable. X has probability distribution x 1 p ( x ) p q , where q = 1- p . • Hence μ = E ( X ) = σ 2 = Var ( X ) = σ = SD ( X ) = . lecture 9, Discrete Random Variables II 5/25 The Bernoulli Distribution The Binomial Distribution Examples lecture 9, Discrete Random Variables II The Binomial Distribution The Binomial Distribution A binomial experiment is an experiment satisfying the following conditions: 1 Experiment consists of a fixed number (say, n ) of trials 2 Each trial is a Bernoulli trial 3 Probability of success, p , is constant from trial to trial — The probability of failure, q , is 1- p 4 Trials are independent • Let S be the total number of successes in the n trials, then S is said to be a binomial ( n , p ) random variable • Possible values that S can take are 0 , 1 ,..., n . • Let X i =1 if the i th trial succeeds, and = 0 otherwise. Then X i is a Bernoulli(p) random variable, and S = X 1 + ... + X n . lecture 9, Discrete Random Variables II 6/25 The Bernoulli Distribution The Binomial Distribution Examples lecture 9, Discrete Random Variables II The Binomial Distribution Examples Examples of Binomial Random Variable 1 Number of 1’s in 10 rolls of a die is a random variable. 2 A multiple-choice test contains 25 questions, each has four choices and one correct answer. The number of correct guesses on the test is a random variable. Binomial or not: 3 A couple decides to have children until they have a girl....
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Lecture9 - lecture 9 Discrete Random Variables II 1/25 The...

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