If $X \sim Exp(1)$ and $Y \sim Exp(1)$, prove $(\frac{X}{Y}, Y)$ is continuos without using the change of variables theorem.

Question

I've been thinking about this problem for a while and I'm not sure which way to go.

Let $X$ and $Y$ be two independent random variables with exponential distribution of parameter 1. Let $U = \frac{X}{Y}$. Without using the change of variables theorem, prove that the vector $(U,Y)$ is continuous.

I imagine this involved calculating the conditional expected value or something or other but I'm not really sure which result would imply continuousness.

Any pointer would be greatly appreciated.

Answer 1

The main tool is the "change of variable" theorem which states that if $X$ is continuous with pdf $f_X$, and $g$ is a $C^1$-diffeomorphism, then $Y:=g(X)$ is continuous with pdf $f_Y(x) := f_X(g^{-1}(x))(g^{-1})'(x)$.

This theorem has two parts : continuity+pdf. But anyway, you only need to check that your transformation is $C^1$, bijective with reciprocal function $C^1$.