I am working through the following problem:
Let $X$ and $Y$ be two independent random variables. Given the marginal pdfs shown below, find the cdf of $Y /X$. (Hint: consider two cases, $0 \le w \le 1$ and $1 < w$.)
$1$. $f_X(x) = 1, 0 \le x \le 1$, and $f_Y(y) = 1, 0 \le y \le 1$
$2$. $f_X(x) = 2x, 0 \le x \le 1$, and $f_Y(y) = 2y, 0 \le y \le 1$
I understand that for $W=Y/X$ the pdf = $\int |x|f_X(x)f_Y(wx) dx$ and that $0\leq w \leq1$ if $X \gt Y$ and similarly $W \gt 1$ if $X \lt Y$ so for
$1$. $\int x dx$
$2$. $\int x*(2x)*(2wx) dx$
I cannot figure out what the proper bounds of the integral in each case should be, I just figure I need to break it up into two separate integrals but I am failing at doing so with keep respect to w.
In the most general case, the limits are $-\infty$ to $\infty$. Here, $W$ can take any positive value, so you can take the limits to be 0 to $\infty$. You can also take the general case of $-\infty$ to $\infty$, and you should find that the value of the integrand is zero for negative $w$, thus recovering 0 to $\infty$ as the effective limits. That your integrand doesn't evaluate to zero for negative values should make you question your formulation.