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: $$E[(\bar{X}_N - \mu)^2] = \frac{1}{N^2}\sum_{i=1}^{N} \operatorname{Var}(X_i)$$

  • $\begingroup$ 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 $\endgroup$ Feb 22 '21 at 15:36

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