I have two subtle questions on the definition of conditional probability and its expectation. Here's the thing.
Question 1.
Definition of a conditional probability density function
$f_{X|Y}(x,y)=\frac{f(x,y)}{f_{Y}(y)}$
And if I want to compute below probability:
$P(0<X<1|0<Y<2)$
then I think below equation is one to calculate:
$P(0<X<1|0<Y<2)=\frac{\int_{0}^{1}\int_{0}^{2}f(x,y)dydx}{\int_{0}^{2}f_{Y}(y)dy}$
and come to think of it. How about this:
$P(0<X<1|0<Y<2)=\int_{0}^{1}\int_{0}^{2}f_{X|Y}(x|y)dydx$
But it is easy to show that
$\frac{\int_{0}^{1}\int_{0}^{2}f(x,y)dydx}{\int_{0}^{2}f_{Y}(y)dy} \neq \int_{0}^{1}\int_{0}^{2}f_{X|Y}(x|y)dydx$
Where are the parts that I am missing at all?
Question 2.
The conditional expectation of $X$, given that $Y=y$, is defined for all values of $y$ such that $f_Y(y)>0$, by
$E[X|Y=y]=\int_{-\infty}^{\infty}xf_{X|Y}(x|y)dx$
Here's my question.
Since we are dealing with continuous random variable cases, we've learned that a value of a pdf at a particular point is meaningless since $P(X=a)=0$, where $X$ is a continuous one.
But, as you can see in the definition of conditional expectation, the given condition is $Y=y$ not that $Y$ is in some interval or something like that.
I don't see why it is a valid definition.
p.s. The text book is Introduction to Probability Models 10th Ed. by Sheldon M. Ross.