Consider two random variables $X$ and $Y$. Let $Y=aX$ where $a$ is a constant. What is the joint Probability Density Function(PDF) of $X$ and $Y$?
The joint PDF is given by
\begin{align*}
\frac{\partial^2}{\partial x \partial y}P(X\leq x,Y \leq y)
\end{align*}
where
\begin{align*}
P(X\leq x,Y \leq y) &= P(X\leq x, X\leq y/a) \\
&=
\begin{cases}
P(X\leq x) & \text{, if } x<y/a\\
P(X\leq y/a) & \text{, if } x>y/a
\end{cases}
\end{align*}
In both cases the partial derivative is zero? Am I missing something, are my calculations wrong or does the PDF not exist?
Best Answer
$(X,aX)$ does not have a PDF, i.e. a density wrt the Lebesgue measure on $\mathbb R^2$.
This because the set $B=\{(x,ax)\mid x\in\mathbb R\}$ which satisfies $P((X,aX)\in B)=1$ has Lebesgue measure $0$.
If $f$ should function as a PDF then we get the contradiction:$$1=P((X,aX)\in B)=\int_Bf(x,y)dxdy=0$$