# Solved – Correlation of linear combination of random variables

How do I proceed further? (assuming what I did is correct in the first place)

Contents

The variables are a random sample – i.e. they are independent. Covariance of independent variables is zero, so you can eliminate \$Cov(X_i, X_j)\$ where \$ineq j\$.

From the property of covariance, you obtain: \$Cov(U_m, U_n)=sum_{i=1}^m sum_{j=1}^n a_i a_j Cov(X_i, X_j)= sum_{i=1}^m a_i^2 Cov(X_i, X_i)=sum_{i=1}^m a_i^2 sigma^2\$ (because \$m leq n\$).

For correlation, lets calculate \$Var(S_n), Var(S_m)\$. Variance of sum of i.i.d. R.V.s equals to sum of their variances, hence \$Var(S_n)=sum_{i=1}^n a_i^2 sigma^2\$, and similarly for \$S_m\$. Then:

\$rho(S_n, S_m)=frac{sum_{i=1}^m a_i^2 sigma^2 }{sqrt{sum_{i=1}^n a_i^2 sigma^2}sqrt{sum_{i=1}^m a_i^2 sigma^2}}\$. It can be seen that if \$m=n\$, then correlation is 1.

If \$a_i=frac{1}{n}\$, then \$Cov(U_m, U_n)=frac{m}{n^2}sigma^2\$ and correlation is \$frac{m}{sqrt{mn}}\$.

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