Let $X$ and $Y$ be jointly continuous random variables with $X$ taking values in some $D$. Give an example of a joint density function $f(x,y)$ such that $E(Y|x)=\ln(x)$:
I have tried so many functions but none seem to fit! Is there a trick/method/theorem I am supposed to use?
I know that:
$$\int_c^d\int_a^{b(x)}\, f(x,y)\, dy\, dx=1$$
and
$$E[Y|x]=\int_a^{b(x)}\,y\cdot f_{Y|x}(y)\, dy=\ln x$$ (so I guess that $$f_{Y|x}(y)=\frac{1}{y^2}$$ and $$b(x)=x\, , a=0$$ but then I can't find suitable $f(x,y)$ and $f_X(x)$.)
Also, $$\int_a^{b(x)}f(x,y)\, dy=f_{X}(x)$$ and $$f_{Y|x}(y)=\dfrac{f(x,y)}{f_X(x)}.$$
Please help me!