CDF of a ratio of exponential variables Let $X$ and $Y$ be independent exponential variables with rates $\alpha$ and $\beta$, respectively. Find the CDF of $X/Y$.
I tried out the problem, and wanted to check to see if my answer of: $\frac{\alpha}{ \beta/t + \alpha}$ is correct, where $t$ is the time, which we need in our final answer since we need a cdf.
Can someone verify if this is correct?

Here is a one-line proof.
$$
\mathbb P(X/Y \le t) = \mathbb P(Y \ge X/t) = \mathbb E[\exp(-\beta X/t)] = \text{MGF}_X(-\beta/t) = (1 - (-\beta/t)1/\alpha)^{-1} = \frac{\alpha}{\alpha + \beta/t} = \frac{\alpha t}{\alpha t + \beta}.
$$
N.B.: For the MGF of an exponential variable, see this table.
