The question is to find the Cumulative Distribution Function (cdf), of $W = X/Y$ given that X and Y are independent random variables and their pdfs are

$f_x(x)=1,0\leq x\leq1$ and $f_y(y) = 1 , 0\leq y \leq 1$.

The book gives a hint that says to consider two cases $0\leq w \leq 1$ and $ 1 < w$.

There are formulas for computing the pdf of W and I would assume I just need to integrate that. My main difficulty is understanding the bounds of this piecewise function. Thank you

  • Sketch the $x$-$y$ plane and indicate on it the region where the joint density $f_{X,Y}(x,y)$ of $X$ and $Y$ is nonzero.

  • What is the region of the plane corresponding to the event $\left\{\frac{X}{Y} \leq w\right\}$ where $w$ is some fixed number in $(0,1)$?

  • Find $P\left\{\frac{X}{Y} \leq w\right\}$ by integrating the joint density over the region you found. If you stop and think a bit and look at your sketches a tad more, you might even be able avoid integrations.

  • Repeat for the case $w > 1$. Verfiy that your answer asymptotically approaches $1$ as $w \to \infty$.

Congratulations. You have found the distribution function $F_W(w)$ of the random variable $W = \displaystyle \frac{X}{Y}$ for $w \geq 0$. Differentiate to get the density function.


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