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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)

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

F(x)

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

Mean

$$E[X]=k$$

Variance

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

MGF

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

Where the moments come from

1. If $Z\sim N(0,1)$ then $Z^2$ has MGF $(1-2t{)}^{-1/2}$, which is Gamma($\frac{1}{2}$, $\frac{1}{2}$).
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. $\mathrm{Var}(X)=\frac{k/2}{(1/2{)}^2}=2k$.

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 the build.

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.