W4-slides2 - Lecture 11 Outline and Examples Discrete...

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Lecture 11- Outline and Examples Discrete probability distributions -Geometric (Ross § 4.8.1) -Negative Binomial (Ross § 4.8.2)
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Geometric distribution Example. Rolling a die. Problem 1. Roll a symmetric die until you have rolled a six. What is the chance that this takes three or less rolls?
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Geometric distribution Example. Rolling a die. Problem 1. Roll a symmetric die until you have rolled a six. What is the chance that this takes three or less rolls? Solution. One roll of a die is a Bernoulli trial because it has two outcomes: either you get a six (S) or not (F). Probability of S is p = 1 / 6 on each trial and the trials (rolls) are independent. Let X = the number of rolls required to produce the first six. Then X has geometric distribution with p = 1 / 6. P ( X 3) = P ( X = 1 , or X = 2 , orX = 3) = P ( X = 1) + P ( X = 2) + P ( X = 3) = p + (1 - p ) p + (1 - p ) 2 p = 1 / 6 + (5 / 6)(1 / 6) + (5 / 6) 2 (1 / 6) = (1 / 6)(1 + 5 / 6 + 25 / 36) 0 . 42
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Geometric distribution Problem 2. Suppose you and I take turns at rolling a die, to see who can first roll a six. Suppose I roll first, then you roll, then I roll, and so on,
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