What does it really mean to normalise a function such that it becomes a probability density function? Say we have a nonnegative integrable function $f:\mathbb R\to\mathbb R$. The normalising procedure would be as follows: $$ \int_{-\infty}^\infty c\cdot f(x)\, dx=1. $$ We woud find our $c$ and have density function $p(x)$. Now what is the meaning of $p(x)$? Say we consider a very small interval $\Delta x$, and some $x\in\Delta x$, then I'm guessing $p(x)\Delta x$ represents the probabilty of having a value $x$?
My question comes from a more practical example, which I will give:
Consider exponential decay; we then have $N(t)=N_0e^{-\lambda t}$. If we normalise this function, we get $\lambda e^{-\lambda t}$, which we recognise as the pdf of the exponential distribution. But what exactly is exponentially distributed? If for instance we consider the mean, we get a value which has to do with time, and not with the size of the population. So it seems like time is distributed, instead of our population...?
In the same way it seems like the input values $x$ are distributed, instead of the actual values of $f(x)$? I see that we can recognise our original function as a scalar multiple of a density function, but what is the practical/physical meaning behind that?
Edit
Oh, maybe I see it partly. We make the area under the curve 1, such that $p(x)dx$ will sort of give us the fraction of the area, which we can interpret as some probability.