Find the distribution of $\bar{X}/\bar{Y}$, i.e. the distribution of the ratio of the sample means of two independent random samples from the (same) exponential distribution with parameter $\lambda>0$.
So far I've played around with potential similarities between the exponential, chi-squared, and Fisher-Snedecors' probability density functions but have not yet found a concrete path for a solution. Any suggestions?
First notice that the $\lambda$ in the numerator and the denominator cancels out, so we may as well assume the distribution of each observation is $e^{-x/2}(dx/2)$ for $x>0.$ This makes the expected value of each observation equal to $2.$ The reason we do that is that the exponential distribution with expected value $2$ is the same as the chi-square distribution with $2$ degrees of freedom. If the respective sample sizes are $m$ and $n,$ we then have a ratio of two independent random variables, each of which is the quotient of a chi-square random variable by half its degrees of freedom. And those two factors of one-half also cancel.
Therefore $\left.\overline X\right/\overline Y \sim F_{2m,2n}.$