Probability Density Function of Chi-Squared Distribution

Definition

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

Let $\Img X = \hointr 0 \infty$.

Let $r$ be a strictly positive integer.

Let $X$ have a $\chi$-squared distribution with $r$ degrees of freedom.

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

$\map {f_X} x = \dfrac 1 {2^{r / 2} \map \Gamma {r / 2} } x^{\paren {r / 2} - 1} e^{-x / 2}$

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: Chi Square Distribution: $39.6$
• 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

• Weisstein, Eric W. "Chi-Squared Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Chi-SquaredDistribution.html