Gamma distribution Probabilities

Calculator of gamma distribution : probabilities, density function (PDF) and cumulative density function (CDF).

Gamma Distribution formulas


X : a random variable with a Gamma distribution
`alpha` : shape parameter (> 0)
`beta` : scale parameter (> 0)

Probability Density Function (PDF)

For x > 0,

`f(x) = 1/(Gamma(alpha))*1/beta^alpha*x^(alpha-1)e^(-x/beta)`
`Gamma(x)` is the gamma function.

Cumulative distribution function (CDF)

`F(x) = \int_-oo^x f(t)\ dt`

`F(x) = 1/(Gamma(alpha))*1/beta^alpha*\int_-oo^xt^(alpha-1)e^(-t/beta)\ dt`


Probability that X is greater than a :
`P(X > a) = 1 - F(a)`

Probability that X is less than b :
`P(X < b) = F(b)`

Probability that X lies between a and b :
`P(a < X < b) = F(b) - F(a)`

k and `theta` parameters

Gamma distribution can also be described by two parameters that replace `alpha` and `beta`:
- k : shape parameter
- `theta` : rate parameter

In this case, above formulas apply and the calculator can be used with,

`alpha = k`

`beta = 1/theta`

See also

Inverse Gamma distribution Calculator
Statistics Calculators