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Let $X_1, X_2, ..., X_n$ be normally distributed with mean $\mu$ and variance $\sigma ^2$, then $\frac{(n-1) S^2}{ \sigma^2}$ has a Chi-Square distribution with $n-1$ degrees of freedom.

How come it's chi-square distributed?

Attempt:

$S^2 = \frac{1}{n-1} \sum(X_i-\overline X)^2 = \frac{1}{n-1}(\sum X_i ^2 - (\sum X_i)^2)$.

Here, the first term $\sum X_i^2$ is chi-squared. But you also need to subtract the second term (which is normal distributed with mean $\mu$, and variance $\frac{\sigma^2}{n}$. How does it make it chi-square distributed?

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The first proof people generally encounter involves MGFs. Note,

$$ \frac{(n-1)S^2}{\sigma^2} = \sum\limits_i^n\left(\frac{X_i-\bar{X}}{\sigma}\right)^2 = \sum\limits_i^n\left(\frac{X_i-\mu}{\sigma}\right)^2 - \left(\frac{\bar{X}-\mu}{\sigma/\sqrt{n}}\right)^2$$

Observe that the first term on the RHS is a chi-square RV with $n$ degrees of freedom. The second term is a chi-square with $1$ degree of freedom. Rearranging,

$$ \frac{(n-1)S^2}{\sigma^2} + \left(\frac{\bar{X}-\mu}{\sigma/\sqrt{n}}\right)^2 = \sum\limits_i^n\left(\frac{X_i-\mu}{\sigma}\right)^2 $$

Now we can find the MGFs both sides,

$$ (1-2t)^{-1/2}M(t) = (1-2t)^{-n/2} \Longrightarrow M(t) = (1-2t)^{-n/2+1/2} = (1-2t)^{-1/2(n-1)}$$

where $M(t)$ is the MGF of $\frac{(n-1)S^2}{\sigma^2}$. Thus, the MGF of $\frac{(n-1)S^2}{\sigma^2}$ is that of a chi-square RV with $n-1$ degrees of freedom.

Note: The above used the identity that $M_{X+Y}(t) = M_X(t)M_Y(t)$ where $X$ and $Y$ are independent. This detail was skipped, but one also needs to show that $S^2$ and $\bar{X}$ are independent which is true in the case of a Normally distributed sample.

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  • $\begingroup$ thanks for your answer. May I ask how did you get from here to here? $$\sum\limits_i^n\left(\frac{X_i-\bar{X}}{\sigma}\right)^2 = \sum\limits_i^n\left(\frac{X_i-\mu}{\sigma}\right)^2 - \left(\frac{\bar{X}-\mu}{\sigma/\sqrt{n}}\right)^2$$ $\endgroup$ – kou Aug 6 '17 at 23:54
  • $\begingroup$ Ignoring the $\sigma^2$, note that $(X_i - \bar{X})^2 = (X_i - \mu + \mu - \bar{X})^2 = (X_i - \mu)^2 - 2(X_i - \mu)(\mu - \bar{X}) + (\bar{X}-\mu)^2$. Taking sums results in $\sum\limits_i^n(X_i - \bar{X})^2 = \sum\limits_i^n(X_i - \mu)^2 - 2n(\bar{X} -\mu)^2 + n(\bar{X}-\mu)^2$. $\endgroup$ – Flowsnake Aug 7 '17 at 0:09
  • $\begingroup$ $$ \frac{(n-1)S^2}{\sigma^2} + \left(\frac{\bar{X}-\mu}{\sigma/\sqrt{n}}\right)^2 $$ Your argument is at best incomplete if it does not mention that these two terms are independent. And their independence is not instantly obvious; that should be explained as well. A substantial step in showing that is to show that $\operatorname{cov}\big( \,\overline X, X_i-\overline X\,\big) = 0. \qquad$ $\endgroup$ – Michael Hardy Nov 29 '19 at 18:07

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