Hi, i posted this in the urgent maths help forum, but i thought it would be better here, any sort of help would be greatly appreciated, thanks.
An estimator $\displaystyle \Phi$ of a parameter $\displaystyle \theta$ is unbiased iff:
$\displaystyle E(\Phi)=\theta$
You first problem is to determine the value of $\displaystyle c$ so that:
$\displaystyle \overline{\sigma^2}=c\sum_{i=1}^{n-1}(X_{i+1}-X_i)$
is an unbiased estimator for the variance.
This is imposible unless the variance is $\displaystyle 0$ as the expectation of the right hand side is $\displaystyle 0$.
Now if instead it was meant to be:
$\displaystyle \overline{\sigma^2}=c\sum_{i=1}^{n-1}(X_{i+1}-X_i)^2$
we have:
$\displaystyle E(\overline{\sigma^2})=c\sum_{i=1}^{n-1}E[(X_{i+1}-X_i)^2]=c (n-1)E(X_{2}^2-2X_{2}X_1+X_1^2)$
This last bit of simplification because the $\displaystyle X_i$ are independently identically distributed.
So:
$\displaystyle E(\overline{\sigma^2})=c (n-1)(\overline{X^2}-2\bar{X}^2+\overline{X^2})=2c(n-1)\sigma^2$
so can you now tell us what value of $\displaystyle c$ will make $\displaystyle \overline{\sigma^2}$ an unbiased estimator?
RonL