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I know how to find the expectation of a function of a Random Variables, I was just wondering that does expectation of a joint distribution exists?

I think, since expectation is an average which by definition means a statistic: a single value that describes a distribution so can we capture the behaviour of the entire distribution in both x and y in a single number, shouldn't we need 2 values for it(one for x and the other for y)?

We can do so for a function of in x and y because that function outputs a single value hence the Expectation is a single number.

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  • $\begingroup$ Indeed, you can find the expected value of the vector $(X, Y)$ (or more generally $(X_1, X_2, \ldots, X_n)$). Or else you can find the expected value of a function $\phi(X, Y)$ [for example, of $XY$ or $[X - E(X)][Y - E(Y)]$, the latter being the covariance of $X$ and $Y$]. Is there any meaningful function which maps $(X, Y)$ to a single random variable that can be said to represent the "value" of $(X, Y)$, and whose expectation (a single real number) can meaningfully be said to be the expected value of $(X, Y)$? Not really. $\endgroup$
    – M. Vinay
    Mar 23, 2019 at 16:08

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You're right. It makes sense to talk about the expectation of any PDF $f(x)$ on any $\mathbb R^n$. The definition is the same as for real-valued distributions. $$ \mathbb E X = \int _{\mathbb R^n} xf(x) dx$$

Here $x$ is a vector and $f(x)$ is a scalar. So we can integrate the vector $x f(x)$ with respect to the normal uniform measure on $\mathbb R^n$ to get a vector in $\mathbb R^n$.

Given two real-valued variables $X$ and $Y$ we define the joint CDF

$$F_{X,Y}(x,y) = P(X < x, Y < y)$$

and the joint PDF

$$f_{X,Y}(x,y) = \frac{\partial^2 F_{X,Y}(x,y)}{\partial x \partial y}$$

The expectation of $(X,Y)$ is defined as the expectation of that PDF:

$$ \mathbb E (X,Y) = \int _{\mathbb R^2} (x,y)f_{X,Y}(x,y) dx dy$$

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  • $\begingroup$ So you're saying that expectation of a joint PDF exists and is a single real no.? searched for "Expectation of a joint PDF" but couldn't find any, all I was getting is the expectation of a function of a x and y. Please give the formulae for finding the expectation of a a joint distribution. $\endgroup$ Mar 23, 2019 at 17:30
  • $\begingroup$ Nope, the expectation is a point in $\mathbb R^2$. $\endgroup$
    – Daron
    Mar 23, 2019 at 22:38
  • $\begingroup$ There is not in general a nice formula for the joint CDF. But if a nice formula exists you can compute the expectation of the joint distribution as described in the first formula. $\endgroup$
    – Daron
    Mar 23, 2019 at 22:39

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