Question

Random variable X follows Ngaa2), (σ < 0 ). A sample X1, X2, X3, randomly. Given the sample mean is: Xn (n 2 2) is chosen 2n 2n 1-1 If, ー1 Find E(Y)

Answer 1

Assume the samples are independent.

Given $Y=\sum_{i=1}^{n}\left ( X_i+X_{n+i}-2\overline{X} \right )^2$ . Now

n+i

The expectation is

i-1 に1 に1

Now,

E (X)Var (X) E(X)

E (X

E(G-).inn (2me..) _ n..j" i-1

Thus,

$E\left (Y \right )=2n\left ( \sigma ^2+\mu ^2 \right )+2\sigma ^2+4n\mu ^2+2n\mu ^2-8n\mu ^2-4\sigma ^2\\ {\color{Blue} E\left (Y \right )=2\left ( n-1 \right )\sigma ^2}$