Relationship between Negative Binomial and Geometric distribution 음이항분포 기하분포

Geometric distribution VS Negative Binomial distribution

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The relationship between Geometric distribution and Negative Binomial distribution is similar to the relationship between Exponential distribution and Gamma distribution.

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The random variable of a negative binomial distribution is the sum of independent and identically distributed geometric distribution.

In other words, Geometric distribution is a special case of Negative Binomial distribution when the number of success(r) equals 1.

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Geometric Distribution

 

when X follows negative binomial distribution, X~Geom(p)

X: the number of independent trials until to get a success.

p: probability to get a success

So geometric distribution is a discrete distribution because it deals with the discrete numbers

 

 

PMF(Probability Mass Function) of Geometric Distribution

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Expected value and Variance of Geometric distribution

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Negative Binomial Distribution

 

when X follows negative binomial distribution, X~Neg.Bin(r,p)

X: the number of independent trials until to get 'r'th success.

r: the number of successes in 'X' trials

p: probability to get a success

So negative binomial distribution is discrete distribution because it deals with the discrete numbers.

 

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PMF(Probability Mass Function) of Negative Binomial Distribution

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Expected value and Variance of Negative Binomial Distribution

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Proof of relations using pdf, E(X) and Var(X)

 

Geom(p) is a special case of NB(r,p) when r(number of success) is 1.

So when substituting r in negative NB(r,p) with 1, it became geometric distribution.

 

 

 

NB(r,p) is a sum of 'r' independent and identically(iid) distributed Geom(p).

So expected value of NB(r,p) is the sum of r expected value of Geom(p).

Also, the variance of NB(r,p) is the sum of r variance of Geom(p).

 

 

 

 

Proof of relations using Moment Generating Function.

This is direct proof using moment generating function. Moment generating function is unique. Moment generating function of Negative Binomial distribution looks like multiplying moment generating function of exponential distribution r times.

 

 

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Proof of relations using Graphs (R)

The graph of NB(r,1) looks the same as Geom(p).

 

 

As NB(r,p) is sum of independent and identically distributed Geom(p), it follows central limit theorem when r is large enough. So graph of NB(r,p) when r is large enough looks symmetric with its mean as it converges to normal distribution.