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Chi-square distribution

A sum of k squared independent standard normals; the sampling distribution of a variance.

Parameters and support

$k$

degrees of freedom - $k > 0$

Support

$x > 0$

The formulas

f(x)

$$f(x) = \dfrac{x^{k/2-1}e^{-x/2}}{2^{k/2}\Gamma(k/2)}$$

F(x)

$$F(x) = \dfrac{\gamma(k/2,\, x/2)}{\Gamma(k/2)}$$

Mean

$$E[X] = k$$

Variance

$$\operatorname{Var}(X) = 2k$$

MGF

$$M(t) = (1-2t)^{-k/2}, \quad t < \tfrac{1}{2}$$

Where the moments come from

1. If $Z \sim N(0,1)$ then $Z^2$ has MGF $(1-2t)^{-1/2}$, which is Gamma($\tfrac12$, $\tfrac12$).
2. Summing $k$ independent copies multiplies the MGFs: $(1-2t)^{-k/2}$.
3. That is the Gamma($\alpha = k/2$, $\lambda = 1/2$) MGF, so $E[X] = \frac{k/2}{1/2} = k$.
4. $\operatorname{Var}(X) = \frac{k/2}{(1/2)^2} = 2k$.

Worked example

The mean, variance, CDF and moment generating function above are re-derived numerically from this distribution’s own density on every test run - summed over the support for a discrete distribution, integrated by quadrature for a continuous one - and compared with the closed forms printed here. A typo on this page fails that test run.

Traps

• Using variance k instead of 2k.
• Forgetting that the chi-square is a special gamma, and re-deriving results that already exist.

Related

• Gamma distribution
• Normal distribution

Drill this: the Exam P question bank has original questions on this distribution, and the recall trainer builds its prompts from exactly the formulas above.