Chapter 3 - Lecture 6 Hypergeometric and Negative Binomial D. Distributions
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1 Chapter 3 - Lecture 6 and Distributions October 14th, 2009 Chapter 3 - Lecture 6 and D
2 experiment random variable distribution Moments Experiment Random Variable Distribution Moments and moment generating functions Geometric Chapter 3 - Lecture 6 and D
3 experiment random variable distribution Moments Definition A hypergeometric experiment is one that satisfies: There is a population of N elements Each element can be characterized as a success or a failure We select a sample of n elements without replacement Chapter 3 - Lecture 6 and D
4 experiment random variable distribution Moments Definition In a hypergeometric experiment the random variable X = number of successes in the sample is called hypergeometric random variable Example: In a large box there are 20 white and 15 black balls. I randomly select 5 balls. Let X =number of white balls in the sample. Then X is a hypergeometric random variable. Chapter 3 - Lecture 6 and D
5 experiment random variable distribution Moments Distribution parameters The Distribution depends on three parameters: The sample size n The population size N The number of successes M in the population Chapter 3 - Lecture 6 and D
6 experiment random variable distribution Moments distribution function If X (N, M, n) then: ( )( ) M N M x n x P(X = x) = ( N n ), max(0, n N + M) x 0, otherwise min(n, M) Chapter 3 - Lecture 6 and D
7 experiment random variable distribution Moments Expected value and variance If X (N, M, n) then: E(X ) = n M ( N ) N n var(x ) = n M ( 1 M ) N 1 N N Chapter 3 - Lecture 6 and D
8 Experiment Experiment Random Variable Distribution Moments and moment generating functions Geometric A negative binomial experiment is one that satisfies: The experiment consists of a sequence of independent trials There are two outcomes of each trial a success and a failure The probability of success is constant from trial to trial and denoted with p. The experiment continues until we observe a total of r successes Chapter 3 - Lecture 6 and D
9 Experiment Random Variable Distribution Moments and moment generating functions Geometric Random Variable In a negative binomial experiment the random variable X = number of failures before we get the r th success is called negative binomial random variable. Example: In a large box there are 20 white and 15 black balls. I randomly select balls until I get 5 white balls. Let X = number of black balls drawn until I get 5 white balls. Then X is a negative binomial random variable. Chapter 3 - Lecture 6 and D
10 Parameters Experiment Random Variable Distribution Moments and moment generating functions Geometric The Distribution depends on two parameters: The number of successes, r, that we want to achieve The probability of success, p. Chapter 3 - Lecture 6 and D
11 distribution function Experiment Random Variable Distribution Moments and moment generating functions Geometric If X NB(r, p) then: ( ) x + r 1 p r (1 p) x, x = 0, 1, 2,... P(X = x) = r 1 0, otherwise Chapter 3 - Lecture 6 and D
12 Expected value, Variance and mgf Experiment Random Variable Distribution Moments and moment generating functions Geometric r(1 p) E(X ) = p r(1 p) var(x ) = p 2 p r M X (t) = (1 e t (1 p)) r Chapter 3 - Lecture 6 and D
13 Experiment Random Variable Distribution Moments and moment generating functions Geometric Geometric distribution When X NB(1, p) then we can say that X is called geometric random variable and follows a geometric distribution All the formulas for the geometric distribution follows from the negative binomial formulas Chapter 3 - Lecture 6 and D
14 Section 3.6 page , 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92 Chapter 3 - Lecture 6 and D
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