## Gamma Distribution Calculator

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

Gamma Distribution Calculator | |
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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) | |

### How to use Gamma Distribution Calculator?

Step 1 - Enter the location parameter (alpha)

Step 2 - Enter the Scale parameter (beta)

Step 3 - Enter the Value of x

Step 4 - Click on “Calculate” button to calculate gamma distribution

Step 5 - Calculate Probability Density

Step 6 - Calculate Probability X less x

Step 7 - Calculate Probability X greater than 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

### $E(X)= \dfrac{\beta}{\alpha}$

## Variance of Gamma distribution

The variance of gamma random variable is

### $V(X) = \dfrac{\beta}{\alpha^2}$.

## Harmonic Mean of Gamma Distribution

The harmonic mean of gamma random variable is

### $H=\frac{\beta-1}{\alpha}$.

## Mode of Gamma distribution

The mode of gamma random variable is

### $\dfrac{\beta-1}{\alpha}$.

## M.G.F. of Gamma Distribution

The moment generating function of gamma random variable is

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

I hope you like **Gamma Distribution Calculator**. Click on Theory to read more about Gamma distribution,graph of gamma distribution,M.G.F and C.G.F of gamma distribution.