The probability density function for Student's -distribution is given by
(1)
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Now define
(2)
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then the cumulative distribution function is given by
(3)
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The mean is 0, so the moments are
(4)
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(5)
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(6)
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(7)
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The mean, variance, skewness, and kurtosis excess are
(8)
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(9)
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(10)
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(11)
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The characteristic function is
(12)
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where is a modified Bessel function of the second kind.
Letting
(13)
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where is the sample mean and is the population mean gives Student's t-distribution.