# Expected value in a linear combination

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 $$EX_2=EX_2'$$.