Variance of Chi-Squared Distribution

Theorem

Let $n$ be a strictly positive integer.

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

Then the variance of $X$ is given by:

$\var X = 2 n$

Proof

By Variance as Expectation of Square minus Square of Expectation, we have:

$\var X = \expect {X^2} - \paren {\expect X}^2$

By Expectation of Chi-Squared Distribution, we have:

$\expect X = n$

We also have:

\(\ds \expect {X^2}\)

\(=\)

\(\ds \prod_{k \mathop = 0}^1 \paren {n + 2 k}\)

Raw Moment of Chi-Squared Distribution

\(\ds \)

\(=\)

\(\ds n \paren {n + 2}\)

\(\ds \)

\(=\)

\(\ds n^2 + 2 n\)

So:

\(\ds \var X\)

\(=\)

\(\ds n^2 + 2 n - n^2\)

\(\ds \)

\(=\)

\(\ds 2 n\)

$\blacksquare$

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