# Product of 2 random variables:domain of integration

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

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 $f_X\left(\frac{z}{y}\right)=0$.

• thank you. I am just looking for the domain of integration; could you please tell me how you get this interval ? – tam Jul 7 '14 at 19:10
• I have moved the explanation from a comment to the answer. – André Nicolas Jul 7 '14 at 19:16
• Could I ask you one more question plz (not related to this topic)? – tam Jul 7 '14 at 19:46
• Sure. But answering questions properly in a comment can be difficult: poor editing facilities, limited space. If it really is a different question, it may be best to ask it separately, and possibly get ideas from several people. – André Nicolas Jul 7 '14 at 20:06
• I had asked this question on mathoverflow, here is the link – tam Jul 7 '14 at 20:16