Student t-distribution arises when estimating the mean of a normally distributed population in situations where the sample size is small and population deviation is unknown. William Sealy Gosset developed the distribution in the beginning of XX century, who published his works under the pseudonym Student.
Probability density function
Probability density function has the following form:
where n - is degrees of freedom and - Gamma function
Cumulative distribution function
Cumulative distribution function can be expressed using Gamma and hypergeometric function:
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-quantile Student is a number which conforms to , where Fn - Student-t cumulative distribution function.
Inverse cumulative distribution function (quantile function) doesn't have simple form, commonly we use pre-calculated values from the tables published by Gosset and other researchers.
The following calculator approximates quantile function value with aid of jStat statistics package: