# Gamma Distribution Model in Mathematics

Introduction :

Suppose an event can occur several times within a given unit of time. When the total number of occurrences of the event is unknown, we can think of it as a random variable. Now, if this random variable X has gamma distribution, then its probability density function is given as follows.

only when x > 0, Î± >0, Î² >0. Otherwise f(x) = 0

where, Î“(Î±) is the value of the gamma function, defined by :

Integrating it by parts, we get that :

for Î± > 1

Thus, Î“(Î±) = (Î±-1)! When Î± is a positive integer.

Represented as –

X ~ GAM(Î², Î±)

Expected Value :

The Expected Value of the Poisson distribution can be found by summing up products of Values with their respective probabilities.

After putting y = x/Î², we get –

Now, after using the identity, Î“(Î± + 1) = Î± Â· Î“(Î±), we get –

Î¼ = Î± Î²

Variance and Standard Deviation :

The Variance of the Gamma distribution can be found using the Variance Formula.

After putting y = x/Î², we get –

But, Î“(Î± + 2) = (Î±+1) Â· Î“(Î±+1) and Î“(Î±+1) =  Î± Â· Î“(Î±)

=> Î“(Î± + 2) = Î±.(Î±+1).Î“(Î±), we get –

Standard Deviation is given by –

Note –

In special case if Î±  = 1, we get exponential distribution with

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