# Weighted sum of Uncorrelated Variance expectation of variances

Assume that we get to observe N random variables$$(X_1, . . . ,X_N)$$ and that each such $$X_i$$ has finite variance. Two questions:

1. If the random variables are uncorrelated. What is $$\operatorname{Var}(\sum_{i=1} w_i X_i)$$ ?

2. Assume that for all $$X_i$$, $$E[X_i]=\mu$$ and that the random variables are uncorrelated, show that: $$E[(\bar{X}_N - \mu)^2] = \frac{1}{N^2}\sum_{i=1}^{N} \operatorname{Var}(X_i)$$

• Welcome to MSE ! For the first question, look up the additive properties of variance. For the second one, you should compute $E[\bar X_N]$ and deduce that $E[(\bar{X}_N - \mu)^2]$ is the variance of $\bar X_N$, you should then be able to conclude Feb 22 '21 at 15:36