The Student’s t distribution
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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.
Distribution name

(Standard) Student’s
t distribution

Common notation


Parameters

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

Domain


Probability density
function


Cumulative distribution
function

where

Mean


Variance


Skewness


(Excess) kurtosis


Characteristic function

where is a Bessel function

Other comments

It is a special case of the generalised hyperbolic
distribution.
Its noncentral moments if is
even and are:
If is even and then , if is
odd and then and if is
odd and then is undefined.

Nematrian web functions
Functions relating to the above distribution may be accessed
via the Nematrian
web function library by using a DistributionName of “student's t”.
Functions relating to a generalised version of this distribution including
additional location (i.e. shift) and scale parameters may be accessed by using
a DistributionName of “student's t3”, see also including
additional shift and scale parameters. For details of other supported
probability distributions see here.
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