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Enterprise Risk Management Formula Book Appendix A.2: Probability Distributions: Continuous (univariate) distributions (a) Normal, uniform, chi-squared

Distribution name

Normal distribution

Common notation

Parameters

= scale parameter ()

= location parameter

Domain

Probability density function

Cumulative distribution function

Mean

Variance

Skewness

(Excess) kurtosis

Characteristic function

The normal distribution is also called the Gaussian distribution. The unit normal (or standard normal) distribution is .

The inverse unit normal distribution function (i.e. its quantile function) is commonly written  (also in some texts  and the unit normal density function is commonly written .  is also called the probit function.

The error function distribution is , where  is now an inverse scale parameter .

The median and mode of a normal distribution are .

The truncated first moments of  are:

where  and  are the pdf and cdf of the unit normal distribution respectively.

The mean excess function of a standard normal distribution is thus

The central moments of the normal distribution are:

 Distribution name Uniform distribution Common notation Parameters = boundary parameters () Domain Probability density function Cumulative distribution function Mean Variance Skewness (Excess) kurtosis Characteristic function Other comments Its non-central moments ( are . Its median is .

 Distribution name Chi-squared distribution Common notation Parameters = degrees of freedom (positive integer) Domain Probability density function Cumulative distribution function Mean Variance Skewness (Excess) kurtosis Characteristic function Other comments Its median is approximately . Its mode is . Is also known as the central chi-squared distribution (when there is a need to contrast it with the noncentral chi-squared distribution).   In the special case of  the cumulative distribution function simplifies to .   The chi-squared distribution with  degrees of freedom is the distribution of a sum of the squares of  independent standard normal random variables. A consequence is that the sum of independent chi-squared variables is also chi-squared distributed. It is widely used in hypothesis testing, goodness of fit analysis or in constructing confidence intervals. It is a special case of the gamma distribution.   As ,  and