I am trying to clarify a fundamental concept regarding probability and density functions. Please bear with me if it is silly.
Say, I need to define the likelihood of two people meeting (random variable $X$) just based on the distance ($y$). And if we assume that they are likely to meet as the distance between them decreases.
Can I say that the probability $p$ that they meet is given by, $p=e^{-y^2}$ ? If yes, why? For my doubt is that this does not qualify to be a density function.
This is indeed not a probability density function, but you don’t want a probability density function; you want a probability. It’s perfectly possible that the probability of the two people meeting is $\mathrm e^{-y^2}$. A probability density function would be required if the distance were a random variable, e.g. if they were to meet at a random distance. Then indeed $\mathrm e^{-y^2}$ would not be a suitable probability density function for the distance $y$, since it’s not normalized. A suitable probability density function for $y$ might be $\frac2{\sqrt\pi}\mathrm e^{-y^2}\mathbb 1[y\ge0]$ (where $\mathbb 1[y\ge0]$ is the indicator function that is $1$ for $y\ge0$ and $0$ otherwise).