# Pareto distribution Probabilities

Calculator of Pareto distribution of parameters `x_m` (scale) and `alpha` (shape).

## Pareto Distribution formulas

X : a random variable following a Pareto distribution

`x_m` : scale parameter (> 0)

`alpha` : shape parameter (> 0)

### Probability Density Function (PDF)

For x greater than `x_m` (inclusive),

`f(x) = (alpha*x_m^alpha)/x^(alpha+1)`

Pour x less than `x_m`,
`f(x) = 0`

### Cumulative distribution function (CDF)

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

`F(x) = 1-(x_m/x)^alpha` for `x>=x_m`

`F(x) = 1` for `x < x_m`

### Probabilities

a is a real number greater than `x_m`.

Probability that X is greater than a :

`P(X > a) = 1 - F(a) = (x_m/a)^alpha`

Probability that X is less than a :

`P(X < a) = F(a) = 1-(x_m/a)^alpha`

## See also

Inverse Pareto distribution

Pareto distribution Chart

Statistics Calculators