# IMSL_NORMALCDF

The IMSL_NORMALCDF function evaluates the standard normal (Gaussian) distribution function. Using a keyword, the inverse of the standard normal (Gaussian) distribution can be evaluated.

Note: This routine requires an IDL Analyst license. For more information, contact your Exelis VIS sales or technical support representative.

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

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 IMSL_NORMALCDF is the distribution function) has mean of zero and variance of 1. The probability that a normal random variable with mean µ and variance σ2 is less than y is given by IMSL_NORMALCDF evaluated at (y – µ)/σ.

The function Φ(x) is evaluated by use of the complementary error function, IMSL_ERFC. The relationship follows below:

If the keyword INVERSE is specified, the IMSL_NORMALCDF function evaluates the inverse of the distribution function, Φ, of a standard normal (Gaussian) random variable; that is:

IMSL_NORMALCDF (x, /INVERSE) = Φ–1 (x)

where:

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 has a mean of zero and a variance of 1.

The IMSL_NORMALCDF function is evaluated by use of minimax rational-function approximations for the inverse of the error function. General descriptions of these approximations are given in Hart et al. (1968) and Strecok (1968). The rational functions used in IMSL_NORMALCDF are described by Kinnucan and Kuki (1968).

## Examples

### Example 1

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.

x1 = (90-100)/15.

p = IMSL_NORMALCDF(x1)

PM, p, Title = 'The probability that X is less than 90 is:'

The probability that X is less than 90 is: 0.252493

x1 = (105 - 100)/15.

x2 = (110 - 100)/15.

p = IMSL_NORMALCDF(x2) - IMSL_NORMALCDF(x1)

PM, p, Title = 'The probability that X is between 105 and ', $

'110 is:'

The probability that X is between 105 and 110 is: 0.116949

## Syntax

Result = IMSL_NORMALCDF(x [, /DOUBLE] [, /INVERSE])

## Return Value

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

## Arguments

### x

Expression for which the normal distribution function is to be evaluated.

## Keywords

### DOUBLE

If present and nonzero, double precision is used.

### INVERSE

If present and nonzero, evaluates the inverse of the standard normal (Gaussian) distribution function. If INVERSE is specified, then argument x represents the probability for which the inverse of the normal distribution function is to be evaluated. In this case, x must be in the open interval (0.0, 1.0).

## Version History

6.4 | Introduced

## See Also