Independent zero mean random variables (find variance of)

Question

Let $X_1, X_2, . . .$ be independent zero mean random variables such that the variance of $X_j$ is $σ_j^2$, which is finite for every $j$. In addition, the (Pearson) correlation coefficient of $X_j$ and $X_k$ is the same $r$ for every $1 ≤ j < k$ integers. (a) Find the variance of $$\frac{X_1}{σ_1} + \frac{X_2}{σ_2} + ... + \frac{X_n}{σ_n}, (n ≥ 1)$$ (b) Prove that $r ≥ 0$.

Solution: $r=\frac{cov(X_j,X_k)}{σ_jσ_k}$, also $Var(\frac{X_1}{σ_1} + \frac{X_2}{σ_2} + ... + \frac{X_n}{σ_n})$ = $Var(\frac{X_1}{σ_1}) + Var(\frac{X_2}{σ_2}) + ... + Var(\frac{X_n}{σ_n})$, and $E(X_jX_k)=0$

And I dont know what to do the next?

Answer 1

Can the solution be: $$Var\left(\frac{X_1}{σ_1}+\frac{X_2}{σ_2}+...+\frac{X_n}{σ_n}\right)=Var\left(\frac{1}{σ_j}\sum_{j=1}^n(X_j)\right)=1$$

Answer 2

Without making any assumptions about independence we have

\begin{eqnarray*} Var\Big[\sum_{i=1}^n\frac{X_i}{\sigma_i}\Big] &=& \sum_{i=1}^n Var\Big[\frac{X_i}{\sigma_i}\Big] + \sum_{i,j=1,i\neq j}^n Cov\Big[\frac{X_i}{\sigma_i},\frac{X_j}{\sigma_j}\Big]\\ &=& \sum_{i=1}^n \frac{1}{\sigma_j^2}Var[X_i] + \sum_{i,j=1, i\neq j}^n \frac{1}{\sigma_i\sigma_j}Cov[X_i,X_j]\\ &=& \sum_{i=1}^n \frac{\sigma_j^2}{\sigma_j^2} + \sum_{i,j=1, i\neq j}^n \frac{1}{\sigma_i\sigma_j}Cov[X_i,X_j]\\ &=& n + \sum_{i,j=1, i\neq j}^n \frac{1}{\sigma_i\sigma_j}Cov[X_i,X_j]\\ \end{eqnarray*}

(a) If the $X_i$ are independent then $Cov[X_i,X_j]=0$ for $i\neq j$ and hence \begin{eqnarray*} Var\Big[\sum_{i=1}^n\frac{X_i}{\sigma_i}\Big] &=& n \end{eqnarray*}

(b) If the $X_i$ are independent then $r_{ij}=\frac{Cov[X_i,X_j]}{\sigma_i\sigma_j}=0$ for $i\neq j$.