lognormalCdf

Evaluates the lognormal cumulative distribution function (CDF).

Synopsis

lognormalCdf(x, amu, sigma)

Required Arguments

float x (Input)

Argument for which the lognormal CDF is to be evaluated. x must be non-negative.

float amu (Input)

Location parameter of the lognormal CDF.

float sigma (Input)

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

Return Value

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

Description

The function lognormalCdf evaluates the lognormal cumulative distribution function (CDF), defined as

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

where

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

is the standard normal CDF.

Example

In this example, we evaluate the CDF at x = 0.7137, amu = 0.0, sigma = 0.5.

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

x = 0.7137
amu = 0.0
sigma = 0.5

p = lognormalCdf(x, 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 %6.4f" % (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 0.2500