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I have a random variable Y, that is defined by:

$$Y = aX_1 + bX_2$$

Where we know $X_1 $ and $X_2$ are independent. How do I write out $EX_1$ and $EX_2$ in terms of only a, b, EY, and VarY?

I have already written out the expectation of Y out following properties of expectation:

$$EY = aEX_1 + bEX_2$$

and to obtain $EX_1$ I can write $\frac{EY - bEX_2}{a}$; however, how would I get rid of the $EX_2$? Essentially, I can write out the expectations of linear combinations, but I have no idea how to write out each individual expectation within that linear combination without using the other variables in that combination.

Finally, how could I solve for the expectation of $X_i$ for a general form $Y = a_1X_1 + ... + a_iX_i$ only in terms of $a_{1,...,i}$, EY, and VarY?

Thank you!

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You can't. Namely, if $X_1\sim\mathcal N(2,1)$, $X_2\sim \mathcal N(-2,1)$, $X'_1\sim\mathcal N(1,1)$ and $X'_2\sim N(-1,1)$ (and, of course, they are independent), then $X_1+X_2, X_1'+X_2'\sim \mathcal N(0,2)$.

Yet, $EX_1,EX_2,EX'_1,EX'_2$ are four distinct numbers whereas your claim would imply $EX_1=EX_1'$ and $EX_2=EX_2'$.

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