How is X/Y distributed when X and Y are uniformly distributed on [0,1]? Let $X$,$Y$ be uniformly distributed continuous random variables on [0,1]. How is the random variable $X/Y$ distributed? 

Step 1. Let $A_k$ be the area of the portion of the square $[0,1]\times [0,1]$ for which $y\leq k x$. 
If $k\in[0,1]$, $A_k$ is the area of a right triangle with its perpendicular sides having lengths $1$ and $k$. If $k\geq 1$, $A_k$ is the area of the square minus the area of a right triangle with its perpendicular sides having length $1$ and $\frac{1}{k}$.
Step 2. Let $Z=\frac{X}{Y}$. $Z$ is obviously distributed over $\mathbb{R}^+$, and for any $k\in\mathbb{R}^+$:
$$ \mathbb{P}\left[\frac{X}{Y}\leq k\right] = A_k = \left\{\begin{array}{rcl}\frac{k}{2}&\text{if}& 0< k\leq 1\\ 1-\frac{1}{2k}&\text{if}& k\geq 1.\end{array}\right.$$
Step 3. By differentiating the previously computed CDF, we have that the probability density function of $Z$, say $f_Z(t)$, is distributed over $\mathbb{R}^+$, equals $\frac{1}{2}$ over $(0,1]$ and $\frac{1}{2t^2}$ over $[1,+\infty)$.
