I have the following expression and I am having trouble understanding it
$$\frac{\int y f_{Y,X}(y,x) dy}{\int y f_Y(y) dy}$$
Where $f_{Y,X}(y,x)$ is the joint density function of random variables $X$ and $Y$ and $f_Y(y)$ the marginal density function of $Y$. For example if $Y$ is income and $X$ is some characteristic, I think this expression can be interpreted as the share of income (over total income) of the population with characteristic $X=x$. My reasoning comes from understanding that in the expression
$$\mathbb{E}(Y|X=x) = \frac{1}{f_X(x)}\int y f_{Y,X}(y,x) dy$$
$f_X(x)$ is the mass of people with characteristic $X=x$ and $\int y f_{Y,X}(y,x) dy$ is the sum (integral) of their incomes. In the case of $\int y f_Y(y) dy$ we are dividing by $1$ since the mass of the total population is one. Also, with discrete random variables it seems to make sense.
What makes me suspicious is the following example. Suppose $X_1 \sim U[0,1]$, $X_2 \sim U[0,1]$ and $Y = b_1 X_1 + b_2 X_2$. Let also $\mu = \int y f_Y(y) dy$. Also $f_{X_2}(X_2)$ is the pdf of $X_2$. Note
$$\frac{1}{\mu}\int y f_{Y,X_2}(y,X_2) dy = \frac{\mathbb{E}(Y|X_2)f_{X_2}(X_2)}{\mu} = \frac{\frac{b_1 + 2b_2 X_2}{2}}{\frac{b_1 + b_2}{2}}\mathbb{I}(0 \leq X_2 \leq 1) = \frac{b_1 + 2b_2 X_2}{b_1 + b_2}\mathbb{I}(0 \leq X_2 \leq 1)$$
Which is not necessarilly between zero and one ($X_2=1$ for example). So is my interpretation wrong? Or im I missing something in the example? If the interpretation is wrong is there a similar expression which can be interpreted as a share?