Input
Formulas and References
For a Student t distribution with degrees of freedom ν:
E[X] = 0 (when ν > 1),
Var(X) = ν/(ν − 2) (when ν > 2).
The distribution is symmetric about zero. Lower degrees of freedom yield heavier tails which affect existence of moments.
FAQs
1) What input does this tool require?
Only degrees of freedom ν. Enter any positive real number to evaluate moment existence and values.
2) Why is the mean undefined for small ν?
The first moment exists only when ν > 1. For ν ≤ 1 the expected value diverges so the mean does not exist.
3) Why can the variance be infinite?
For 1 < ν ≤ 2 the second moment diverges even though the mean exists. Heavy tails cause infinite variance.
4) How precise are the numeric results?
Results are computed using double precision arithmetic. Displayed values reflect the precision of your input.
5) What happens for very large ν?
The distribution approaches the normal distribution. Variance tends toward 1 and the mean remains 0.
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