Specific case of the ratio of random variables

Question

I have an odd question related to the ratio of two random variables. Given random variables $$X \sim G(\mu_1, \sigma_1^2) \;\;\; and\;\;\; Y \sim G(\mu_2,\sigma_2^2)$$ there are a good number of resources that talk about the resulting distribution for $Z = \frac{X}{Y}$ (in particular when X and Y are independent). My question is how do you talk about this when $Y=X$? Then we have $Z= \frac{X}{X}$. In my ignorance I would say $Z$ comes from a distribution that gives the value 1 almost always.

Answer 1

Your intuition is exactly correct. One way to derive the result formally would be to look at the cumulative distribution function of Z $F_z(t)$.

$$F_Z(t) = P(Z \leq t) = P(X/Y \leq t) = P(1 \leq t) = \begin{cases} 0, & t < 1 \\ 1, & t \geq 1 \end{cases}$$

From this you can then see that the distribution of $Z$ is simply a point mass of 1 at 1.

I should note that the solution above is the general way one approaches solving the distribution for a random variable that is some transformation of other variables. In this case we don't need any of that of course, we can just say that $Z = X/X = 1$, i.e. Z is a constant.