# Gamma distribution Probabilities

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

## Gamma Distribution formulas

### Notations

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

### Probabilities

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