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I have three random variables: $X$, $Y$, and $Z$ and I know their expected values. I want to calculate:

$$E\left[\frac{3-X-Y}{Z}\right].$$

I know that $E\left[\frac{1}{X}\right] \neq \frac{1}{E\left[ X\right]}$ but how can one approach a problem like that?

bajun65537
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    It depends on the details. What do you know about the joint distribution of $(X,Y,Z)$? If you know nothing, then there's nothing you can do to compute the expectation, because it could be (literally) anything, including undefined or infinite. – whuber Jul 31 '22 at 15:36
  • What if it is assumed that the joint distribution of $(X,Y,Z)$ is normal? – bajun65537 Aug 03 '22 at 14:24
  • Then, unless it is a degenerate Normal distribution, the fraction can be expressed as a sum of which one term is a ratio of independent Normally distributed variables and https://stats.stackexchange.com/questions/299722 shows the expectation is undefined. – whuber Aug 03 '22 at 14:27

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