Probability of two variables of having the same value

Question

Let $X$ and $Y$ be two random variables, whose PDFs $f_X$ and $f_Y$ are uniform. $f_X$ and $f_Y$ may overlap. For instance, they could represent two score distributions for two tuples $x$ and $y$ in a database.

Which is the probability for $X$ and $Y$ of having the same value $v$ (e.g., for the tuples $x$ and $y$ of having the same score)?

I have tried with an integral, but this returns me back $0$ as a result, since it is evaluated on an interval that is a point (the value $v$), and an integral that is evaluated on a point returns $0$ as a result.

Answer 1

Since $X, Y$ are continuous it is indeed the case that $\text{Prob}(X=Y)=0$.

Answer 2

Yes you are right. If you set X and Y to be continuous r.v.'s, then the probability of getting the exact same value is 0 (because the integral of the density function from a point to the same point is always 0).

You could try an approximation by stating a range instead of a number. For example, you can ask what is the probability that both X and Y fall in the interval $(v - \epsilon, v + \epsilon)$, where $\epsilon$ is some small number (such as 0.005).