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!

  • $\begingroup$ To get the Latex to show up, remove the " ` " marks and do not indent the $$ equations. $\endgroup$ – angryavian May 18 at 1:56
  • $\begingroup$ Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. $\endgroup$ – dantopa May 18 at 1:59

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 $EX_2=EX_2'$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.