Sampling Distribution of the Variance of a Normally Distributed Random Sample with an Unknown Expected Value

Topics: Statistics - Normal Distribution Sampling Distribution

---

(theorem)

Let $X_1,\dots ,X_n$ be iid random variables (random sample) with a normal distribution that has an unknown expected value, but a known variance ${\sigma }^2$.

Then:

$$\frac{n-1}{{\sigma }^2}{S}_{n-1}^2\sim {\chi }_{n-1}^2$$

…where:

$$S_{n-1}^2=\frac{{\sum }_{i=1}^n{\left(x_i-\overline{X}\right)}^2}{n-1}$$

(i.e. quasivariance)

Known Expected Value

When we do know the expected value of our random variables, do something similar (this time with the traditional sample variance).