## Gamma Distribution Calculator

Use Gamma Distribution Calculator to find the probability density and cumulative probabilities for Gamma distribution with parameter $\alpha$ and $\beta$.

Gamma Distribution Calculator
Location Parameter $\alpha$:
Scale Parameter $\beta$
Value of x
Results
Probability density : f(x)
Probability X less than x: P(X < x)
Probability X greater than x: P(X > x)

## Definition of Gamma Distribution

A continuous random variable $X$ is said to have a gamma distribution with parameter $\alpha$ and $\beta$ if its p.d.f. is given by $$f(x; \alpha,\beta) = \frac{1}{\beta^\alpha \Gamma(\alpha)} x^{\alpha-1}e^{-x/\beta}; x>0; \alpha, \beta>0$$

The parameter $\alpha$ is the scale parameter and $\beta$ is the shape parameter of gamma distribution.

## Mean of Gamma Distribution

The mean or expected value of gamma random variable is

## Variance of Gamma distribution

The variance of gamma random variable is

## Harmonic Mean of Gamma Distribution

The harmonic mean of gamma random variable is

## Mode of Gamma distribution

The mode of gamma random variable is

## M.G.F. of Gamma Distribution

The moment generating function of gamma random variable is

### $M_X(t)=\bigg(1-\dfrac{t}{\alpha}\bigg)^{-\beta}, t<\alpha$

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