Exercise :
Let $X>0$ and $Y>0$ continuous random variables with joint probability density function (pdf) $f(x,y)$. If $W=Y/X$, then find the pdf $f_W(w)$. What happens to $f_W(w)$ and its expression when $X,Y$ are independent ?
Attempt-Question :
$$F_W(w)=P\{W \leq w\} = P\{Y/X \leq w\}=\int \int_Df(x,y)dxdy$$
$$\Rightarrow$$
$$F_W(w) = \int\bigg[\int f(x,y)dx\bigg]dy$$
It is : $Y/X \leq w \Leftrightarrow Y \leq wX$ since $X>0$. This means that on a $X-Y$ axis system, we'll need every value below $wX$.
$$=\int_{-\infty}^{ωx}\bigg[\int_{-\infty}^{+\infty}f(x,y)dx\bigg]dy$$
Which leads to :
$$f_W(w) = \frac{dF_W(w)}{dw} = \frac{d}{dw}\int_{-\infty}^{ωx}\bigg[\int_{-\infty}^{+\infty}f(x,y)dx\bigg]dy = \int_{-\infty}^\infty\bigg[\frac{d}{dw}\int_{-\infty}^{wx}f(x,y)dy\bigg]dx$$
Question : Is my approach and the integral bounds correct ? How would I take it one step further from then on ?