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I'm having yet another confusion with my probability homework and I would love some help.

3.2.5: $X$ follows Uniform$[3,7]$, $Y$ follows Exponent$(9)$. Find $\mathop{{}\mathbb{E}}(-5X -6Y).$

3.2.6: $X$ follows Uniform$[-12,-9]$, $Y$ follows N$(-8,9)$. Find $\mathop{{}\mathbb{E}}(11X+14Y+3).$

For the life of me, I don't understand how 3.2.5 $\neq 5E(X) + -6E(Y)$, and how 3.2.6 $\neq 11E(X)+14E(Y)+3$. I must be missing some theory here. Is there a theorem that could help me out?

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  • $\begingroup$ ??? what makes you think that those are not equal? $\endgroup$
    – David C. Ullrich
    Commented Nov 9, 2020 at 3:29
  • $\begingroup$ Stupid answer key! I've done this and checked the solution using Michael J. Evans and Jeffrey S. Rosenthal's answer key and I thought I had made some mistake. Now I see that the only mistake made was reading the textbook in the first place. $\endgroup$
    – JerBear
    Commented Nov 9, 2020 at 3:45

1 Answer 1

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The linearity of expectation does not require the random variables to be independent. For any random variables $X$, $Y$ with finite expectation, and for any scalars $a, b, c$, we have $$\operatorname{E}[aX + bY + c] = a \operatorname{E}[X] + b \operatorname{E}[Y] + c.$$

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