In this question, I'm having a hard time understanding how specifically to set the bounds for the CDF.

Let $X$ and $Y$ be independent variables. Find the CDF of $W=Y/X$ using the marginal PDFs

$ f_X(x)=2x, f_Y(y)=2y $ where $0 \le x,y \le 1 $.

To do this, need to consider two different cases where $0 \le w \le 1 $ and $w > 1$.

In the case that $0 \le w \le 1 $, the bounds would be $0$ and $1$:

$f_W(w)=\int_0^1 x(f_X(x))(f_Y(wx)dx $

However in the case that $w > 1$, the bounds are $0$ and $1/w$, as shown here:

$f_W(w)=\int_0^{1/w} x(f_X(x))(f_Y(wx)dx $

Why is this the case? I'm having a hard time understanding how the bounds are set, as I wouldn't have chosen these.

What if the boundaries for the for the marginal PDFs had changed, like to $0 \le x \le 2 $, or maybe $0 \le y \le 2 $; how would this affect the bounds for $f_W(w)$?

Thank you!

  • $\begingroup$ please add the self-study tag. $\endgroup$ – Xi'an Mar 6 '19 at 6:16
  • $\begingroup$ Just added the tag! $\endgroup$ – Sarina Mar 7 '19 at 2:44

When considering the change of variable from $(x,y)$ to $(x,w)$, the constraint $$0\le x\le 1\qquad 0\le y\le 1$$becomes$$0\le x\le 1\qquad 0\le xw\le 1$$which can be rewritten as$$0\le x\le 1\qquad 0\le w\le 1/x$$and as$$0\le w\le \infty\qquad 0\le x\le \min\{1,1/\omega\}$$


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