+25 I have this exercise I'm having trouble with.
Could you help me out?
Suppose we have n independent random variables $$X_1,X_2,X_3,....,X_n$$ with expected value $$E[X_i] = \mu$$ and variance $$V[X_i] = \sigma^2$$ for any i = 1,..., n. Let us also consider the following random variables.
$$Y:= \sum^n_{i=1}a_iX_i$$ and $$Z := \sum^n_{i=1}b_iX_i$$ for known constants $$a_i \mbox{ and } b_i, i=1,...,n$$.
By computing exercise (1-1) I already know $$E(Y) = \mu\sum^n_{i=1}a_i$$
By computing exercise (1-2) I also know $$V(Y) = \sigma^2\sum^n_{i=1}a_i$$
By computing exercise (1-3) I know that if $$a_i = \frac{1}{n} \forall i, E[Y] = \mu \mbox{ and } V[Y] = \frac{\sigma^2}{n}$$
Excerside (1-4) asks me to compute the following;
Compute the Covariance Cov[Y,Z] and Corr[Y,Z].
I dont get much further than
$$E[(\sum^n_{i=1}a_iX_i-\sum^n_{i=1}a_i\mu)(\sum^n_{i=1}b_iX_i-\sum^n_{i=1}b_i\mu)]$$
for the covariance after which things go horribly wrong.
Please help me :)
Rhino
