Product of 2 random variables:domain of integration

Question

I am trying to compute the PDF of the product of two ind. random variables: $Z=XY$, where $0\leq x \leq d$ and $ 0\leq y \leq 1 $. ($0<d<1$)

I found this formula : $ f_Z(z) = \int_{-\infty}^{+\infty} \frac{1}{|y|} f_{Y}(y) f_X( \frac{z}{y}) dy $. How to determine the domain of integration ?

Thanks

Answer 1

For $z$ in the interval $(0,d)$, the integration is over the interval $\frac{z}{d}\le y\le 1$ .

The variable $y$ must be positive and $\le 1$, since $f_Y(y)=0$ outside $[0,1]$.

The restriction $y\ge \frac{z}{d}$ comes from the fact that if $y\lt \frac{z}{d}$, then $\frac{z}{y}\gt \frac{z}{z/d}=d$. But then $f_X\left(\frac{z}{y}\right)=0$.