Minimum Covariance
I have a question that I got stuck on for a while.
Let $Y_1,Y_2,…,Y_n$ be a rendom variable in (Ω,p), for $1\leq i \leq n$ we get that $Y_i$~$Unif\{1,-1\}$.
let $c\in R$ and for any $i\neq j$ let $COV(Y_i,Y_j) = c$.
prove that $c≥-\frac{1}{n-1}$
I know that $COV(x,y)=E[xy]-E[x]E[y]$.
$E[Y_i] = E[Y_j] = 0$.
But I'm not Sure how to approach $E[Y_iY_j]$
Solution 1:
Guide:
With these information, you should be able to get the inequality.
$$Var\left(\sum_{i=1}^n Y_i \right)=\sum_{i=1}^n Var(Y_i)+2\binom{n}{2}c$$ $$E[Y_i^2]=1$$ $$E[Y_i]=0$$
$$Var\left( Y_i\right)=1$$