Confidence interval for normal sample variance

The case of the famous middle [edit]

Let be - independent sampling from the normal distribution, where - known mean. Define arbitrary and build - confidence interval for unknown dispersion .

Statement. Random value

has a distribution . Let be - -quantile of this distribution. Then we have:

.

After substitution of the expression for and simple algebraic transformations we get:

.

The case of the unknown mean [edit]

Let be - independent sampling from the normal distribution, where , - unknown constants. Construct a confidence interval for the unknown variance .

Fisher's theorem for normal samples. Random value

,

Where - unbiased sample variance, has a distribution . Then we have:

.

After substitution of the expression for and simple algebraic transformations we get:

.