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.
![[SmartChart]](I/StudentsTDistribution_files/image001.gif)
![[SmartChart]](I/StudentsTDistribution_files/image002.gif)
![[SmartChart]](I/StudentsTDistribution_files/image003.gif)
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Distribution name
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(Standard) Student’s
t distribution
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Common notation
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Parameters
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 =
degrees of freedom ( , usually is
an integer although in some situations a non-integral can
arise)
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Domain
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Probability density
function
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Cumulative distribution
function
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where 
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Mean
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Variance
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Skewness
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(Excess) kurtosis
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Characteristic function
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where  is a Bessel function
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Other comments
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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.
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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.
NAVIGATION LINKS
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