Distribution name

Lognormal distribution

Common notation

Parameters

 = scale parameter ()

 = location parameter

Domain

Probability density function

Cumulative distribution function

Mean

Variance

Skewness

(Excess) kurtosis

Characteristic function

No simple expression that is not divergent

Other comments

The median of a lognormal distribution is  and its mode is .

The truncated moments of   are:

Distribution name

(Standard) Student’s t distribution

Common notation

Parameters

 = degrees of freedom (, usually  is an integer although in some situations a non-integral  can arise)

Domain

Probability density function

Cumulative distribution function

where

Mean

Variance

Skewness

(Excess) kurtosis

Characteristic function

where  is a Bessel function

Other comments

The Student’s t distribution (more simply the t distribution) arises when estimating the mean of a normally distributed population when sample sizes are small and the population standard deviation is unknown.

It is a special case of the generalised hyperbolic distribution.

Its non-central moments if  is even and  are:

If  is even and  then , if  is odd and  then  and if  is odd and   then  is undefined.