lognormalInverseCdf

Evaluates the inverse of the lognormal cumulative distribution function (CDF).

Synopsis

lognormalInverseCdf(p, amu, sigma)

Required Arguments

float p (Input)

Probability for which the inverse of the lognormal CDF is to be evaluated. p must lie in the closed interval [0, 1].

float amu (Input)

Location parameter of the lognormal CDF.

float sigma (Input)

Shape parameter of the lognormal CDF. sigma must be positive.

Return Value

Function value, the probability that a lognormal random variable takes a value less than or equal to the returned value is the input probability p. A value of NaN is returned if an input value is in error.

Description

The function lognormalInverseCdf evaluates the inverse CDF of a lognormal random variable with location parameter amu and scale parameter sigma. The probability that a standard lognormal random variable takes a value less than or equal to the returned value is p (p=P).

$$P = \frac{1}{\sigma\sqrt{2\pi}} \int_0^x \frac{e^{-\frac{1}{2}\left(\frac{\log(t)-\mu}{\sigma}\right)^2}}{t} dt = \varphi \left(\frac{\log(x) = \mu}{\sigma}\right)$$

where

$$\varphi(y) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{y} e^{-\frac{1}{2} u^2} du$$

In other words

$$F^{-1} = (P|\mu,\sigma) = x$$

Example

In this example, we evaluate the inverse CDF at p = 0.25, amu = 0.0, sigma = 0.5.

from __future__ import print_function
from numpy import *
from pyimsl.stat.lognormalInverseCdf import lognormalInverseCdf

p = 0.25
amu = 0.0
sigma = 0.5

x = lognormalInverseCdf(p, amu, sigma)
print("The probability that lognormal random variable X")
print("with location parameter amu = %3.1f " % amu)
print("and shape parameter sigma = %3.1f " % sigma)
print("is less than or equal to %6.4f is %4.2f" % (x, p))

Output

The probability that lognormal random variable X
with location parameter amu = 0.0 
and shape parameter sigma = 0.5 
is less than or equal to 0.7137 is 0.25