Expectation of Chi-Squared Distribution

Theorem

Let $n$ be a strictly positive integer.

Let $X \sim \chi_n^2$ where $\chi_n^2$ is the chi-squared distribution with $n$ degrees of freedom.

Then the expectation of $X$ is given by:

$\expect X = n$

Proof

\(\ds \expect X\)

\(=\)

\(\ds \prod_{k \mathop = 0}^0 \paren {n + 2 k}\)

Raw Moment of Chi-Squared Distribution

\(\ds \)

\(=\)

\(\ds n\)

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): chi-squared distribution
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): chi-squared distribution
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $13$: Probability distributions
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $15$: Probability distributions