Suppose $X, Y$ are independent uniform(0,1) random variables. For some arbitrary $t$, I want to find $P(X/Y \le t)$. I am trying to draw a picture to figure this out, but I don't think I am accomplishing too much. Could I have some direction on how to do this?
For clarity, $t$ is arbitrary. I have no information on what it is.
First of all note that $T=\frac{X}{Y} \in(0;+\infty)$
The result is the following
$$\mathbb{P}[T\leq t]=\begin{cases} 0, & \text{if $t<0$ } \\ \frac{t}{2}, & \text{if $0\leq t<1$}\\ 1-\frac{1}{2t}, & \text{if $t\geq 1$}\\ \end{cases}$$
To understand this result take a look at the following drawing
$$\mathbb{P}\Bigg[\frac{X}{Y} \leq t\Bigg]=\mathbb{P}\Bigg[Y > \frac{X}{t}\Bigg]$$
thus the probability you are looking for is the purple area. When $t<1$ it is easy to calculate 1 minus white triangle's area