Variance of Chi-Squared Distribution
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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