Finding the density for $\Phi(X,Y)$

Question

Let the random variables $T$, $T'$ and $U$ be independent with $T ∼ \exp(1)$, $T ∼ \exp(1)$ and $U ∼ U(0, 1)$. Let $X = \log\left(\frac U {1-U}\right)$ and $Y = (2U-1)(T + T')$.

Let $G = \{(x, y): xy > 0\}$ and define the mapping $\Phi: G → (0, 1) × (0, ∞)$ by

$$\Phi(x,y) = \left( \frac{e^x}{e^x + 1}, \frac{y(e^x + 1)}{e^x - 1}\right).$$

Find the density of the vector $\Phi(X,Y)$.

Could someone help me with this exercise? How do I start solving it?

Answer 1

Big hint: you can directly show $\Phi(X,Y) = (U, T+T')$ by substituting in the definitions of $X$ and $Y$.