Student's t-Distribution
X ~ t_ν
Heavy-Tailed / Small-Sample Family • Published July 20, 2026
The small-sample fix for the Normal: history from William Sealy Gosset's 1908 Guinness brewery work under the pen name 'Student,' its exact Normal-over-Chi-Square construction, why heavier tails than the Normal account for the extra uncertainty of an estimated standard deviation, convergence to the standard Normal as degrees of freedom grow, why a moment generating function never exists despite finite low-order moments, a verified closed-form Fisher information via digamma and trigamma functions, and simulation via its own defining construction.
This reference sheet requires a free account.
Sign up free to read the full Student's t-Distribution reference sheet.
Sign up freePrefer a saved copy?
About This Reference Sheet
The small-sample fix for the Normal: history from William Sealy Gosset's 1908 Guinness brewery work under the pen name 'Student,' its exact Normal-over-Chi-Square construction, why heavier tails than the Normal account for the extra uncertainty of an estimated standard deviation, convergence to the standard Normal as degrees of freedom grow, why a moment generating function never exists despite finite low-order moments, a verified closed-form Fisher information via digamma and trigamma functions, and simulation via its own defining construction.
Support
x ∈ (-∞, ∞)
Parameters
ν > 0 (degrees of freedom)