Suppose $X_1, X_2, . . . , X_6$ is a random sample of a normal random variable with mean µ and variance $σ^2$ .
Determine c such that c[$(X_1 − X_2)^2$ + $(X_3 − X_4)^2$ + $(X_5 − X_6)^2$] is an unbiased estimator of $σ^2$.
I've attempted to get the second moments of the $(X_i − X_j)$'s and then to use these and solve for c
So far I managed to find $E[X_i − X_j]$ = µ-µ = 0 and $Var[X_i − X_j]$ = $Var[X_i]$ + $Var[X_j]$ - 2$Cov[X_i,X_j]$ = 2$σ^2$- 2$Cov[X_i,X_j]$
Thus $E[(X_i − X_j)^2]$ = 2$σ^2$- 2$Cov[X_i,X_j]$
$E[(X_i − X_j)^2]$ = 2$σ^2$+2$µ^2$ - $E[X_iX_j]$
Does anyone have any hints on how to progress from here, there's no mention in the question that the sample is i.i.d