lecturenotesweek4

lecturenotesweek4 - ISyE 2027 Probability with Applications...

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Unformatted text preview: ISyE 2027 Probability with Applications Polly B. He Fall10, Week 4 Reading: Section 4.3-4.4, 5.1 The Bernoulli & Binomial Distributions Definition: A discrete random variable X has a Bernoulli Distribution with parameter p , p 1, if its probability mass function is given by p X (1) = P ( X = 1) = p and p X (0) = P ( X = 0) = 1- p We denote this distribution by Ber ( p ), and write it as X Ber ( p ). Example 1: We ip a fair coin, = { H, T } . X ( ) = 1 = H = T We see that X Ber ( 1 2 ). Example 2: Define Y = 1 if it rains tomorrow otherwise Note: 1. When does the Bernoulli distribution arise? It often relates to experiments where there are only two outcomes: whether or not, usually encoded as 1 and 0. 2. The Bernoulli distribution is a building block for the next standard distribution, the Binomial Distribution. A motivating example of the Binomial distribution: Let us consider in an exam, there are 10 questions. R i = 1 if the i-th question is answered correctly otherwise 1 Assume that R i Ber ( p ) (a reasonable assumption). Also assume that these 10 ques- tions are physically independent experiments (in reality, you should have some doubts about this assumption)....
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lecturenotesweek4 - ISyE 2027 Probability with Applications...

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