I must have missed something important in the formulation of the problem. Can you please help clarify how should the following simple problem be formulated and where my mistake is.
Let $Z = \frac{Y}{X}$ where $X, Y\sim U(0,1)$ and independent. Find $f_z(z)$, the pdf of $Z$.
$P(Z \leq z) = P(\frac{Y}{X} \leq z) = \iint_A f_{X, Y}(x,y)dx dy $, where $A = \left\{(x,y) = 0\leq x\leq 1, 0\leq y\leq1, \frac{y}{x}\leq z \right\}$
$P(Z \leq z) = F_z(z) = \int_0^1(\int_0^{xz}1\cdot 1dy)dx = \int_0^1 zx dx = z\frac{x^2}{2}|_0^1 = \frac{z}{2}$
The above is only valid for $0 \leq Z \leq1$ or equivalently $Y \leq X $. Where step in my calculation did I introduce $Y \leq X $ ?
