normalCdf¶

Evaluates the standard normal (Gaussian) distribution function.

Synopsis¶

normalCdf (x)

Required Arguments¶

float x (Input): Point at which the normal distribution function is to be evaluated.

Return Value¶

The probability that a normal random variable takes a value less than or equal to x.

Description¶

The function normalCdf evaluates the distribution function, Φ, of a standard normal (Gaussian) random variable; that is,

\[\phi(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{x} e^{-t^2/2} dt\]

The value of the distribution function at the point x is the probability that the random variable takes a value less than or equal to x.

The standard normal distribution (for which normalCdf is the distribution function) has mean of 0 and variance of 1. The probability that a normal random variable with mean μ and variance \(\sigma^2\) is less than y is given by normalCdf evaluated at \((y-\mu)/\sigma\).

\(\Phi(x)\) is evaluated by use of the complementary error function, erfc. The relationship is:

\[\phi(x) = \mathrm{erfc}\left(-x/\sqrt{2.0}\right) / 2\]

Figure 9.20 — Plot of Φ(x)

Example¶

Suppose X is a normal random variable with mean 100 and variance 225. This example finds the probability that X is less than 90 and the probability that X is between 105 and 110.

from __future__ import print_function
from numpy import *
from pyimsl.math.normalCdf import normalCdf

x1 = (90.0 - 100.0) / 15.0
p = normalCdf(x1)
print("The probability that X is less than 90 is %6.4f" % (p))

x1 = (105.0 - 100.0) / 15.0
x2 = (110.0 - 100.0) / 15.0
p = normalCdf(x2) - normalCdf(x1)
print("The probability that X is between 105 and 110 is %6.4f" % (p))

Output¶

The probability that X is less than 90 is 0.2525
The probability that X is between 105 and 110 is 0.1169