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I have two random variables $X$ and $Y$ whose probability distributions are given by $P_X(x)$ and $P_X(x)$ respectively. I know that we can get the mean some function $f(x,y)$ by integrating across the joint probabilty distribution: $$ E[f(x,y)] = \int \int f(x,y) P_{XY}(x,y) \ dx \ dy $$

my question is what happens when we instead have a function of the expected value of one of these parameters, $g(E[X],y)$. would we be able to simply integrate across the remaining random parameter given that $E[X]$ is no longer a random variable?

$$ E[g(E[X],y)] = \int g(E[X],y) P_{Y}(y) \ dy $$

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2 Answers 2

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First some remarks:

Actually you should write $\mathbb E[f(X,Y)]$ and $\mathbb E[g(\mathbb E[X],Y)]$ using capitals for the random variables, as you also introduced them.

Secondly what you call "probability distributions" are in fact "probability density functions".

Thirdly your write $P_{XY}(x,y)$ for the PDF (=probability density function) of random vector $(X,Y)$.

You better write it as $P_{X,Y}(x,y)$ to avoid confusion with $XY$ which is also a random variable and has a PDF.


In the situation you sketch $\mathbb EX$ is a constant.

Denoting it by $\mu$ for a suitable function $g:\mathbb R^2\to\mathbb R$ we get a "new" random variable $Z:=g(\mu,Y)$ and for its expectation we indeed have the equality: $$\mathbb EZ=\mathbb Eg(\mu,Y)=\int g(\mu,y)P_Y(y)dy$$

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Yes you can, but whether it is meaningful or not actually depends upon what you want to obtain. In most cases $$\mathbb E_{X,Y}g(X,Y) \ne g(\mu_X,Y) \ne g(\mu_X,\mu_Y)$$ unless $g$ is linear in $X$ and $Y$ (think about Jensen's inequality).

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