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.
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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.
The x value at which to evaluate (defined when x is greater than mu)
Location parameter that sets the lower edge of the distribution (any real number)
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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Levy Distribution Calculator