Expectation of Student's t-Distribution

Theorem

Let $k$ be a strictly positive integer.

Let $X \sim t_k$ where $t_k$ is the $t$-distribution with $k$ degrees of freedom.

Then the expectation of $X$ is equal to $0$ for $k > 1$, and does not exist otherwise.

Proof

From the definition of the Student's t-Distribution, $X$ has probability density function:

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

with $k$ degrees of freedom for some $k \in \R_{>0}$.

From the definition of the expected value of a continuous random variable:

$\ds \expect X = \int_{-\infty}^\infty x \map {f_X} x \rd x$

So for $k > 1$:

\(\ds \expect X\)

\(=\)

\(\ds \dfrac {\map \Gamma {\frac {k + 1} 2} } {\sqrt {\pi k} \map \Gamma {\frac k 2} } \int_{-\infty}^\infty \dfrac x {\paren {1 + \dfrac {x^2} k}^{\frac {k + 1} 2} } \rd x\)

\(\ds \)

\(=\)

\(\ds \dfrac {\map \Gamma {\frac {k + 1} 2} } {\sqrt {\pi k} \map \Gamma {\frac k 2} } \paren {\int_{-\infty}^0 \dfrac x {\paren {1 + \dfrac {x^2} k}^{\frac {k + 1} 2} } \rd x + \int_0^\infty \dfrac x {\paren {1 + \dfrac {x^2} k}^{\frac {k + 1} 2} } \rd x}\)

Sum of Integrals on Adjacent Intervals for Continuous Functions

\(\ds \)

\(=\)

\(\ds \dfrac {k \map \Gamma {\frac {k + 1} 2} } {2 \sqrt {\pi k} \map \Gamma {\frac k 2} } \paren {\int_\infty^1 \dfrac 1 {\paren u^{\frac {k + 1} 2} } \rd u + \int_1^\infty \dfrac 1 {\paren u^{\frac {k + 1} 2} } \rd u}\)

U-substitution Substituting $u = 1 + \dfrac {x^2} k$ and $\dfrac k 2 \rd u = x \rd x$

\(\ds \)

\(=\)

\(\ds \dfrac {k \map \Gamma {\frac {k + 1} 2} } {2 \sqrt {\pi k} \map \Gamma {\frac k 2} } \paren {-\int_1^\infty \dfrac 1 {\paren u^{\frac {k + 1} 2} } \rd u + \int_1^\infty \dfrac 1 {\paren u^{\frac {k + 1} 2} } \rd u}\)

Reversal of Limits of Definite Integral

\(\ds \)

\(=\)

\(\ds \dfrac {\map \Gamma {\frac {k + 1} 2} } {2 \sqrt {\pi k} \map \Gamma {\frac k 2} } \paren {\dfrac {-2} {k - 1 } } \paren {-\intlimits {\paren u^{-\frac {k - 1} 2} } 1 \infty + \intlimits {\paren u^{-\frac {k - 1} 2} } 1 \infty}\)

Primitive of Power, Fundamental Theorem of Calculus

\(\ds \)

\(=\)

\(\ds 0\)

Definite Integral of Odd Function/Corollary

$\Box$

When $k = 1$, we have the natural log evaluated at infinity where it is undefined.

Hence:

\(\ds \)

\(\)

\(\ds \dfrac {\map \Gamma 1} {2 \sqrt \pi \map \Gamma {\frac 1 2} } \paren {-\int_1^\infty \dfrac 1 u \rd u + \int_1^\infty \dfrac 1 u \rd u}\)

U-substitution Substituting $u = 1 + \dfrac {x^2} k$ and $\dfrac k 2 \rd u = x \rd x$ and Reversal of Limits of Definite Integral

\(\ds \)

\(=\)

\(\ds \dfrac {\map \Gamma 1} {2 \sqrt \pi \map \Gamma {\frac 1 2} } \paren {-\bigintlimits {\map \ln u} 1 \infty + \bigintlimits {\map \ln u} 1 \infty}\)

Fundamental Theorem of Calculus, Definition of Natural Logarithm, Logarithm Tends to Infinity

Therefore, the expectation of $X$ does not exist:

$\blacksquare$

Sources

• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $15$: Probability distributions