Is there some way of understanding more deeply why exactly it is that the probability mass function or probability density functions of various random variables have the shapes that they have when you plot them?
For example, a priori, it is pretty obvious that a continuous, uniformly distributed r.v. will have a PDF that looks like a flat line. However, simply looking at the PMF for a binomial r.v. (i.e. not plotting it), is there any intuition for why it might look the way it does when you plot it, which, for larger and larger $n$, seems to be the shape of a normal distribution?
A priori, either from the equation of the PMF or just thinking about the definition of a binomial r.v., a series of $n$ trials with probability $p$ that they are successful whose PMF $P(X=i)$ is the probability of $i$ successes in those $n$ trials, why would it look anything like the way it does? A kind of normal distribution shape. Why would the probabilities be highest around the middle and drop off towards the ends?