Errata of conditional expectation in page 9

[Whisht]

1. Errata 1

$$ \mathbb{E}[XY] = \int_{X,Y} f_{X,Y}(x,y) dF_X(x)dF_y(y) $$

should be

$$ \mathbb{E}[XY]= \int_{X}\int_Y xy f_{X,Y}(x,y) dx dy $$

2. Errata 2

$$ \mathbb{E}_{\varphi(X,Y)|X=x}[=]\int_{-\infty}^{\infty} \varphi(x,y)f_{Y|X}(y|x)dx $$

should be

$$ \mathbb{E}\left[{\varphi\left(X,Y\right)\mid X=x}\right]=\int_{-\infty}^{\infty} \varphi\left(x,y\right)f_{Y\mid X}\left(y\mid x\right)d\color{red}y $$

The conditional expectation is a random variable of condition $X$.

Meanwhile, I suggest you add another formula:

$$ \mathbb{E}_{XY}\left[\varphi\left(X,Y\right)\right]=\mathbb{E}_X\left[\mathbb{E}\left[\varphi\left(X,Y\right)\mid X\right]\right] $$

[mavam]

Thanks a lot for noting these issues!

I've fixed the 2nd errata already in the main branch.

Regarding the 1st. I'm not exactly sure what it is that you are pointing out. The current version looks slightly different from what you write:

The main fix you're suggesting seems to be the inclusion of $xy$, which is already there. The notation $dF_x(x)$ is Wassermann's way of writing "both discrete and continuous summation". The only other difference I noticed is that you used the marginal integrals instead of the joint. Is that what you are proposing?

Regarding the last suggestion, thanks, I've added it as follows:

[Whisht]

Sorry, I am not familiar with Wassermann's way. I thought $dF(x)$ is differential of $F(x)$, i.e. $dF(x)=f(x)dx$. So the notation you used here makes me think $\mathbb{E}[XY] = \int_{X,Y} xy f_{X,Y}dF_X(x)dF_Y(y) = \int_{X,Y} xy f_{X,Y}f_X(x)f_Y(y) dxdy$. That's why I pointed out the 1st errata. Anyway, It looks OK under Wassermann's way.