Consider two mutually independent random variables ${v_1},{v_2}$, which are the Gaussian white noise with distribution $N(0,1)$. We suppose they satisfy the following equation $({v_1}+a_1){w_1} + ({v_2}+a_2){w_2} = 0$, where ${a_1},{a_2}$ are the given constants, ${w_1},{w_2}$ are some values to be determined. My question is whether $a_1{w_1} +a_2{w_2}$ has the zero expectation? Can anyone help me?
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Do you mean $w_1,w_2$ are values to be determined by the distributions of $v_1,v_2$? – Harry Alli Jan 02 '18 at 02:10
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The description is incomplete. You need to describe $w_1 \ and\ w_2$ better. Are they random variables or unknown constants? What is their relationship, if any, to $v_1\ and \ v_2$? – herb steinberg Jan 02 '18 at 03:06
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If $w_1,w_2$ are values taken on by the distributions $v_1,v_2$, then it becomes a simple case of expectation algebra. Actually in both cases it does. https://en.wikipedia.org/wiki/Algebra_of_random_variables – Harry Alli Jan 02 '18 at 03:48
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Yes, $w_1,w_2$ are values to be determined by the distributions of $v_1,v_2$. For different samples of $v_1,v_2$, we can determine the solution of ${w_1},{w_2}$. Naturally, ${w_1},{w_2}$ are random variables. However, my main concern is whether $a_1{w_1} +a_2{w_2}$ have the zero expectation if $v_1,v_2$ have the zero expectation? – wenbo Jan 02 '18 at 10:25