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$X$ and $Y$ are independent standard uniform random variables. What is the density of $Z = X/Y$?

So far I have:

$f_X(x) = f_Y(y) = 1\text{ if }0 \le x,y \le 1$

$\begin{align} f_Z(z) & = \int_{-\infty}^\infty f_X(zx)f_Y(x)|x| dx \\ & = \int_{-\infty}^\infty f_X(zx)f_Y(x)x dx \\ & = \int_{-\infty}^\infty f_X(zx)x dx \\ \end{align}$

I think that $f_X(zx)$ in the integrand can be replaced with $1$ but how would I change the bounds of the integral?

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    I have no idea how your formula $f_Z(z) = \int_{-\infty}^\infty f_X(zx)f_Y(x)|x| dx$ was obtained. I dislike the use of such mystical magical general formulas, which you presumably found in your textbook or were "taught" in class and so is in your class notes, because students are never able to actually use the formula in any given specific case. It is far better to learn$a$general _method_ such as the one outlined in my previous comment rather than rely on$a$canned formula. Maybe someone else can help you.2012-10-28

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