Levy Distribution Calculator

Compute the probability density, cumulative probability (lower tail), upper tail, and median of the Levy distribution with location mu and scale c, with charts.

Input

For the Levy distribution with location mu and scale c, compute the density, cumulative probability (lower tail), upper tail, and median at a point x.

Evaluation point x

The x value at which to evaluate (defined when x is greater than mu)

Location mu

Location parameter that sets the lower edge of the distribution (any real number)

Scale c

Positive scale parameter controlling the spread

Result

Lower tail probability F(x) up to x = 2

0.47950012

Upper tail 1 minus F(x)

0.52049988

Density f(x)

0.10984782

Median

2.19810934

Mean

Diverges to infinity

Probability density function f(x)

Cumulative distribution function F(x)

How it works

The Levy probability density is f(x) = sqrt(c / 2pi) times exp(-c / (2(x-mu))) divided by (x-mu)^1.5 for x greater than mu, and 0 otherwise.
The cumulative distribution (lower tail) is F(x) = erfc(sqrt(c / (2(x-mu)))), using the complementary error function erfc. The upper tail probability is 1 minus F(x).
The Levy distribution is a stable distribution with a very heavy tail, so both its mean and variance diverge to infinity. The median is mu + c / (2 (inverse erfc of one half) squared), about mu + 2.198 times c, which is finite.
Enter a positive scale c and any real location mu. When x is less than or equal to mu the value is outside the support, so the density and all probabilities are 0.

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