# Beta distribution Probabilities

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

## Beta Distribution formulas

### Notations

X : a random variable following a beta distribution

`alpha` : shape parameter (> 0)

`beta` : second shape parameter (> 0)

### Probability Density Function (PDF)

For x between 0 and 1 (inclusive),

`f(x) = (Gamma(alpha+beta))/(Gamma(alpha)*Gamma(beta))*x^(alpha-1)(1-x)^(beta-1)`

`Gamma(x)` is the gamma function.
### Cumulative distribution function (CDF)

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

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

## See also

Inverse Beta distribution Calculator

Beta distribution Chart

Statistics Calculators