Intuition (and name) of this result about bivariate expected values?

Question

I've seen the following result about expected values in class:

$$E(g(X,Y))=\int_{\mathbb{R}} E(g(X,y)|Y=y)f_Y (y) \ dy $$ Where $ E(g(X,y)|Y=y)= \int_\mathbb{R}g(x,y)f_{X|Y}(x|y) \ dx $.
(I'm not sure if this comes directly from the definition of $E(g(X)|Y=y)?$ Can we just treat the $y$ as a constant?)

I was wondering whether anyone could explain why this is "intuitively true"? Is it to do with the law of total probability? Is there a name for this result?

The reasoning I've tried/heard: "We "fix" a value of $Y=y$, and calculate the expectation given assuming this is true, and then times this by the "density" and sum over all possible $y$ values.

I was a little dissatisfied with this explanation because it seems to just be describing the integral. But, I might have missed something in the explanation which made it less intuitive for me.

Thank you!

Answer 1

Is there a name for this result?

The Law of Total Expectation.$$\mathsf E(Z)=\mathsf E\big(\mathsf E(Z\mid Y)\big)$$

The reasoning I've tried/heard: "We "fix" a value of $Y=y$, and calculate the expectation given assuming this is true, and then times this by the "density" and sum over all possible $y$ values.

I was a little dissatisfied with this explanation because it seems to just be describing the integral.

It seems fairly intuitive.   The expectation of a random variable is its probability weighted average.   If we partition the support of the random variable then the expectation is the probability weighted average of the conditional expectations over each partition. $$\mathsf E(g(X,Y))=\sum_i\mathsf E(g(X,Y)\mid Y\in S_i)~\mathsf P(Y\in S_i)\qquad\text{where }\{S_i\}\text{ partitions the support for Y}$$

If we carefully make the partitions small enough, that summation approaches the integral.$$\mathsf E(g(X,Y))=\int_\Bbb R \mathsf E(g(X,Y)\mid Y=y)\,f_{\small Y}(y)\,\mathrm d y$$

$\tiny\text{[Assuming that conditional expectation is definable and that the marginal pdf exists.]}$

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Well, when $X,Y$ are random variables with a joint probability density function, $f_{\small X,Y}$, then by the definition of expectation we have:

$$\mathsf E(g(X,Y)) =\iint_{\Bbb R^2} g(x,y)\, f_{\small X,Y}(x,y)\,\mathrm d(x,y)$$

Now by definition of conditional probability density, if the marginal and conditional density functions exist (which they usually do): $$\begin{align}\mathsf E(g(X,Y)) &=\iint_{\Bbb R^2} g(x,y)\, f_{\small X\mid Y}(x\mid y)\,f_{\small Y}(y)\,\mathrm d(x,y)\\[2ex]&=\int_\Bbb R\left(\int_\Bbb R g(x,y)\,f_{\small X\mid Y}(x\mid y)\,\mathrm d y\right)\,f_{\small Y}(y)\,\mathrm d x\end{align}$$

So... this gives us a definition for conditional expectation with respect to values of $Y$.

$$\begin{align}\mathsf E(g(X,Y)) &=\int_\Bbb R\mathsf E(g(X,Y)\mid Y{=}y)\,f_{\small Y}(y)\,\mathrm d y\\[2ex]&=\mathsf E\big(\mathsf E(g(X,Y)\mid Y)\big)\end{align}$$