Relationship between Exponential and Gamma distribution 지수분포 감마분포 관계

Exponential Distribution VS Gamma Distribution

 

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

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The random variable of Gamma distribution is the sum of independent and identically distributed Exponential distribution.

As Random variable of Negative Binomial distribution is the sum of iid Geometric distribution.

 

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Exponential Distribution

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When X follows exponential distribution, X~EXP(λ)

X: the time until next event occurs.

So exponential distributon is continuous distribution because it deals with time which is continuous.

 

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PDF(Probability Density Function) and CDF of Exponential Distribution

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Expected value and Variance of Exponential distribution

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Gamma Distribution

When X follows gamma distribution, X~GAM(R, λ)

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X: Time Until Rth event occurs

Sum of R independent and identically (iid) distributed exponential random variable

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PDF(Probability Density Function) of =Gamma Distribution

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when X: Time until Rth event occurs

R: The number of event occurrences.

 

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Expected value and Variance of Gamma distribution

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X~GAM(R, λ )

 

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Gamma distribution with usually used parameters

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There are two notations usually used in Gamma distribution instead of using R and X

I am gonna use the second notation with parameter Alpha and Beta to prove the relationship between exponential and gamma distribution here.

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Proof of relationship using pdf, E(x) and Var(x).

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There are several reasons why I can say Gamma Distribution and Exponential distribution are highly related.

Firstly, I can give you the reason by substituting 1 with alpha of gamma distribution, you can see that the distribution becomes having the same distribution with exponential distribution with parameter Beta.

 

Also, expected value and variance of the sum of exponential distribution became expected value and variance of gamma distribution with parameter (R, lambda) where R is the total number in sum of exponential distributions and lambda is the same parameter of the exponential distribution.

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Proof of relationship using moment generating function

This is direct proof using the moment generating function. Moment generating function is unique. Moment generating function of gamma distribution looks like multiplying moment generating function of exponential distribution n times.

 

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Proof of relationship Using graphs

Graph using R

The graph of gamma distribution with shape parameter 1 is and scale parameter 1.5 looks the same with exponential distribution with parameter 1.5

In other words, gamma distribution with shape parameter 1 became exponential distribution.

As gamma distribution is sum of independent and identically distributed exponential distribution, it follows the central limit theorem when shape parameter is large enough. So graph of gamma distribution with large shape parameter looks symmetric of its mean as it converges to normal distribution.