Question

compute a 95% upper confidence interval of σ^2 .

Q1: Given xi,T2, , and sample second moment is2. Compute the sample variance ,Xn. Suppose n = 10, sample first moment (i.e., sample mean) is s Σία 1x2-2. Compute the sample variance.

Answer 1

The sample variance, $s^2$ = $E(X^2)$ - $E^2(X)$ = $\frac{\sum_{i = 1}^{n} x_i^2}{n}$ - $\overline{x}^2$ = 2 - 1 = 1.

We know, $\frac{(n-1)s^2}{\sigma^2}$ ~ $\chi^2_{n-1}$

The 95% upper confidence interval of $\sigma^2$ is = [0, $\frac{(n-1)s^2}{\chi^2_{0.025}}$ ], where, n = 10, $s^2$ = 1

= [0, 3.33]. (Ans).