Question

4.1.5. F , x, let (S)2=(1/n)? 1(x-?Find or a random sample X1, EKS')2]. Compare this with EG). i-1

Answer 1

By definition, $\small (S')^2$ becomes the variance of the Random Sample X. Where variance is the expectation of the squared deviation of a random variable from its mean.

${\displaystyle {\begin{aligned}\operatorname {Var} (X)&=\operatorname {E} \left[(X-\operatorname {E} [X])^{2}\right]\\&=\operatorname {E} \left[X^{2}-2X\operatorname {E} [X]+\operatorname {E} [X]^{2}\right]\\&=\operatorname {E} \left[X^{2}\right]-2\operatorname {E} [X]\operatorname {E} [X]+\operatorname {E} [X]^{2}\\&=\operatorname {E} \left[X^{2}\right]-\operatorname {E} [X]^{2}\end{aligned}}}$

$\small S' = E[X^2] - E[X]^2$

$\small E[S'] = E[E[X^2]] - E[E[X]^2]$

Now,

Since the expected value of an expected value is just that. It stops being random once you take one expected value, so iteration doesn't change.

So,

$\small E[S' ]= E[X^2] - E[X]^2$

Also $\small S^2$ is the sample variance with the formula:

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

So, By comparision,

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

$\small S^2 = \frac{n}{n-1}(S')^2$

So,

$\small E[S^2] = \frac{n}{n-1}E[(S')^2]$

So, $\small E[S^2]$    is $\small \frac{n}{n-1}$ times $\small E[(S')^2]$ .