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.

f(x) = \frac{1}{\beta^\alpha Γ(\alpha)} x^{(\alpha - 1)}e^{\frac{-x}{\beta}}

only when x > 0, α >0, β >0. Otherwise f(x) = 0

where, Γ(α) is the value of the gamma function, defined by :

Γ(α) = \int^{\infty}_{0} x^{(\alpha - 1)}e^{-x}\,dx

Integrating it by parts, we get that :

Γ(α) = (α-1)Γ(α-1)    
for α > 1

Thus, Γ(α) = (α-1)! When α is a positive integer.

Represented as -

X ~ GAM(β, α)

Expected Value :

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

\mu = E(X) = \int^{\infty}_{-\infty} x.f(x) dx
\mu = \frac{1}{\beta^\alpha Γ(\alpha)} \int^{\infty}_{0}x. x^{(\alpha - 1)}e^{\frac{-x}{\beta}} dx

After putting y = x/β, we get -

\mu = \frac{\beta}{Γ(\alpha)} \int^{\infty}_{0} y^{\alpha} e^{-y} dy = \frac{\beta Γ(\alpha+1)}{Γ(\alpha)}

Now, after using the identity, Γ(α + 1) = α · Γ(α), we get -

μ = α β

Variance and Standard Deviation :

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

σ^2 = E( X − μ )^2 = E( X^2 ) − μ^2
E(X^2) = \int^{\infty}_{-\infty} x^2.f(x) dx
E(X^2) = \frac{1}{\beta^\alpha Γ(\alpha)} \int^{\infty}_{0}x^2. x^{(\alpha - 1)}e^{\frac{-x}{\beta}} dx

After putting y = x/β, we get -

E(X^2) = \frac{\beta^2}{Γ(\alpha)} \int^{\infty}_{0} y^{\alpha+1} e^{-y} dy\\ = \frac{\beta^2 Γ(\alpha+2)}{Γ(\alpha)}

But, Γ(α + 2) = (α+1) · Γ(α+1) and Γ(α+1) =  α · Γ(α)

=> Γ(α + 2) = α.(α+1).Γ(α), we get -

E(X^2) = \beta^2.\alpha.(\alpha+1)
So, Var(X) = E(X^2) - \mu^2
 Var(X) = \beta^2.\alpha.(\alpha+1) - \alpha^2\beta^2
Var(X) = \sigma^2 = \alpha\beta^2

Standard Deviation is given by -

\sigma = \sqrt{\alpha \beta^2} = \beta\sqrt{\alpha}

Note -

In special case if α  = 1, we get exponential distribution with \mu = \beta\\ \sigma^2 = \beta^2