My original problem statement:
Let $x$ be a random variable with pdf $f_x(x)$ and let y be $y=x$.
(a) Find the joint pdf $f_{x,y}(x,y)$.
(b) Find the conditional pdfs $f_{x|y}(x|y), f_{y|x}(y|x)$
I've already given some thoughts how the joint pdf should behave:
Picturing the joint pdf as $f_{x,y} = f{y|x}*f(x)$ and assuming a fixed $x$ the result should be 0 for $x \neq y$ and some other value for $x = y$. But I have really a hard time getting the idea which value that might be.
Any hint how this problem can be approached is highly appreciated.
Best Answer
The random couple $(X,Y)$ has no PDF since its support is included in the diagonal $\{(x,x)\mid x\in\mathbb R\}$, which has Lebesgue measure zero.
Likewise, the conditional distribution of $X$ conditionally on $Y$ (or vice versa) has no density. For every $y$, the conditional distribution of $X$ conditionally on $Y=y$ is $\delta_y$ the Dirac mass at $y$. This measure has no density with respect to the Lebesgue measure.
Edit: One might add that the only reason why distributions are useful is to compute means. That is, the distribution of a random element $Z$ allows to compute $E[u(Z)]$ for every measurable real-valued function such that the expectation exists. Here, the distribution of $(X,Y)$ is crystal clear since, considering the distribution $P_X$ of $X$, one has $$ E[u(X,Y)]=\int u(x,x)\mathrm dP_X(x). $$ For example, if $P_X$ has a density $f_X$, $$ E[u(X,Y)]=\int u(x,x)f_X(x)\mathrm dx. $$