Probability Density Function of Student's t-Distribution

Definition

Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.

Let $X$ have a Student's $t$-distribution with $k$ degrees of freedom.

Then the probability density function of $X$ is given by:

$\map {f_X} x = \dfrac {\map \Gamma {\frac {k + 1} 2} } {\sqrt {\pi k} \map \Gamma {\frac k 2} } \paren {1 + \dfrac {x^2} k}^{-\frac {k + 1} 2}$

where $\Gamma$ denotes the gamma function.

Proof

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Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 39$: Probability Distributions: Student's $t$ Distribution: $39.5$
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $15$: Probability distributions

• Weisstein, Eric W. "Student's t-Distribution.." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Studentst-Distribution.html