1. ## Expected Value

For a sample of identically and independently distributed variables , each having a mean μ and variance $\displaystyle \sigma^2$

for $\displaystyle \subscript{i}=1, 2, ...., n$

what is
i) $\displaystyle E(X_i \bar{X}), where \bar{X}= (\sum{X_i})/n$

ii) $\displaystyle E(\bar{X}^2)$

iii) $\displaystyle E[\bar{X}(X_i-\bar{X})]$

thanks a lot

2. Originally Posted by cky78
For a sample of identically and independently distributed variables , each having a mean μ and variance $\displaystyle \sigma^2$

for $\displaystyle \subscript{i}=1, 2, ...., n$

what is
i) $\displaystyle E(X_i \bar{X}), where \bar{X}= (\sum{X_i})/n$

ii) $\displaystyle E(\bar{X}^2)$

iii) $\displaystyle E[\bar{X}(X_i-\bar{X})]$

thanks a lot
Because $\displaystyle X_i, \ X_j , \ i\ne j$ are independent $\displaystyle E(X_iX_j)=\mu^2$ (just write out the expectation of the product and observe that independence means that $\displaystyle p(x_i,x_j)=p(x_i)p(x_j)$)

Then the rest should follow easily

(you might also need $\displaystyle E(X_iX_i)=\sigma^2+\mu^2$ )

CB