Levy Distribution Percent Point Calculator

Find the percent point (quantile) of the Levy distribution from probability p, location and scale using a closed form. Supports lower and upper tail.

Input

Enter probability p, location mu and scale c to compute the percent point (quantile) of the Levy distribution in closed form.

Probability p

A value greater than 0 and less than 1 (e.g. 0.95)

Probability type

Treat p as the lower tail or the upper tail probability

Location mu

Location parameter where the support begins

Scale c

Positive scale parameter controlling the spread

Percent point x with lower tail probability p = 0.95

254.31444455

Location mu

Scale c

Median

2.19810934

Lower probability F(x)

0.95

Upper probability 1 − F(x)

0.05

Density f(x)

9.817472e-5

Mean and variance

Diverge (undefined)

Probability density f(x)

Cumulative distribution F(x)

The Levy cumulative distribution is F(x) = erfc( sqrt(c / (2(x − mu))) ) for x greater than mu. Its density is f(x) = sqrt(c / 2pi) · exp(−c / (2(x − mu))) / (x − mu)^1.5.
For a lower probability p the percent point has the closed form x = mu + c / (2 · (erfinv(1 − p))²), where erfinv is the inverse error function. A rational initial guess is refined with Newton iterations for high accuracy.
When the upper tail is selected, p is replaced by 1 − p before applying the same formula. The Levy distribution has a very long right tail, and both its mean and variance diverge and are undefined.
The median equals mu + c / (2 · (erfinv(0.5))²) ≈ mu + 2.198 c. Enter a probability p strictly between 0 and 1, and a positive scale c.

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