Negative Binomial Distribution

[Pages:71]Introduction to the Negative Binomial

Distribution

Negative Binomial Distribution in R

Relationship with

MGF, Expected Value

Geometric distribution and Variance

Relationship with other Thanks! distributions

Negative Binomial Distribution

Andre Archer, Ayoub Belemlih, Peace Madimutsa

Macalester College

November 30, 2016

1/??

Introduction to the Negative Binomial

Distribution

Negative Binomial Distribution in R

Relationship with

MGF, Expected Value

Geometric distribution and Variance

Relationship with other Thanks! distributions

Outline

1 Introduction to the Negative Binomial Distribution Defining the Negative Binomial Distribution Example 1 Example 2: The Banach Match Problem Transformation of Pdf Why so Negative? CDF of X

2 Negative Binomial Distribution in R R Code Example 3

3 Relationship with Geometric distribution 4 MGF, Expected Value and Variance

Moment Generating Function Expected Value and Variance 5 Relationship with other distributions Possion Distribution

2/??

Introduction to the Negative Binomial

Distribution

Negative Binomial Distribution in R

Relationship with

MGF, Expected Value

Geometric distribution and Variance

Relationship with other Thanks! distributions

Introduction to the Negative Binomial Distribution

Introduction to the Negative Binomial Distribution

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Introduction to the Negative Binomial

Distribution

Negative Binomial Distribution in R

Relationship with

MGF, Expected Value

Geometric distribution and Variance

Defining the Negative Binomial Distribution

Relationship with other Thanks! distributions

X NB(r , p) Given a sequence of r Bernoulli trials with probability of success p, X follows a negative binomial distribution if X = k is the number of trials needed to get to the rth success.

Pdf of X P(X = k) = k - 1 pr (1 - p)k-r r -1

where X = r , r + 1, ? ? ?

4/??

Introduction to the Negative Binomial

Distribution

Negative Binomial Distribution in R

Relationship with

MGF, Expected Value

Geometric distribution and Variance

Defining the Negative Binomial Distribution

Relationship with other Thanks! distributions

Pdf of X P(X = k) = k - 1 pr (1 - p)k-r r -1

where X = r , r + 1, ? ? ?

P(X = k) = P(rth on kth trial)

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Introduction to the Negative Binomial

Distribution

Negative Binomial Distribution in R

Relationship with

MGF, Expected Value

Geometric distribution and Variance

Defining the Negative Binomial Distribution

Relationship with other Thanks! distributions

Pdf of X P(X = k) = k - 1 pr (1 - p)k-r r -1

where X = r , r + 1, ? ? ?

P(X = k) = P(rth on kth trial) = P(r-1th on k-1 trials) ? P(success on kth trial)

6/??

Introduction to the Negative Binomial

Distribution

Negative Binomial Distribution in R

Relationship with

MGF, Expected Value

Geometric distribution and Variance

Defining the Negative Binomial Distribution

Relationship with other Thanks! distributions

Pdf of X P(X = k) = k - 1 pr (1 - p)k-r r -1

where X = r , r + 1, ? ? ?

P(X = k) = P(rth on kth trial) = P(r-1th on k-1 trials) ? P(success on kth trial) = k - 1 pr-1(1 - p)k-1-(r-1) ? p r -1

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Introduction to the Negative Binomial

Distribution

Negative Binomial Distribution in R

Relationship with

MGF, Expected Value

Geometric distribution and Variance

Defining the Negative Binomial Distribution

Relationship with other Thanks! distributions

Pdf of X P(X = k) = k - 1 pr (1 - p)k-r r -1

where X = r , r + 1, ? ? ?

P(X = k) = P(rth on kth trial) = P(r-1th on k-1 trials) ? P(success on kth trial) = k - 1 pr-1(1 - p)k-1-(r-1) ? p r -1 = k - 1 pr-1(1 - p)k-r ? p r -1

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