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PMF: The possible values of a Negative Binomial random variable are x = r,r + 1,... and the probability that the rth success occurs on the xth trial is File Preview nb(x;r,p) = FACT: If X1,..., X, are independent Geometric(p) random variables, then r X = Σ Xi Σ X; ~ Negative Binomial (r,p) i=1 PROPOSITION: The expected value and variance of a Negative Binomial random variable with parameters r and p are E(X) = r 1 - p V(X) =r " Ρ p² Example: You roll a fair six-sided die until you see the third six. What is the probability that this happens on the tenth roll?

PMF: The possible values of a Negative Binomial random variable are x = r,r + 1,... and the probability that the rth success occurs on the xth trial is File Preview nb(x;r,p) = FACT: If X1,..., X, are independent Geometric(p) random variables, then r X = Σ Xi Σ X; ~ Negative Binomial (r,p) i=1 PROPOSITION: The expected value and variance of a Negative Binomial random variable with parameters r and p are E(X) = r 1 - p V(X) =r " Ρ p² Example: You roll a fair six-sided die until you see the third six. What is the probability that this happens on the tenth roll?

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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PMF: The possible values of a Negative Binomial random variable are x = r,r + 1,... and the probability that the rth success occurs on the xth trial is File Preview nb(x;r,p) = FACT: If X1,..., X, are independent Geometric(p) random variables, then r X = Σ Xi Σ X; ~ Negative Binomial (r,p) i=1 PROPOSITION: The expected value and variance of a Negative Binomial random variable with parameters r and p are E(X) = r 1 - p V(X) =r " Ρ p² Example: You roll a fair six-sided die until you see the third six. What is the probability that this happens on the tenth roll?
Transcribed Image Text:PMF: The possible values of a Negative Binomial random variable are x = r,r + 1,... and the probability that the rth success occurs on the xth trial is File Preview nb(x;r,p) = FACT: If X1,..., X, are independent Geometric(p) random variables, then r X = Σ Xi Σ X; ~ Negative Binomial (r,p) i=1 PROPOSITION: The expected value and variance of a Negative Binomial random variable with parameters r and p are E(X) = r 1 - p V(X) =r " Ρ p² Example: You roll a fair six-sided die until you see the third six. What is the probability that this happens on the tenth roll?
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