Sure, given a real valued random variable, it has a cumulative distribution. If that distribution is differentiable, then the random variable also has a density

Sure, given a real valued random variable, it has a cumulative distribution. If that distribution is differentiable, then the random variable also has a density (distribution). Fine. Alas, that doesn't mean that in practice we should try to find what the distribution is! Actually, in practice, it's generally difficult to know with much accuracy what a distribution is. Then, for the OP, in practice it's much more difficult to know much about the tails or when they are fat or not or if fat how fat. The OP wants to claim that the normal distribution applies to heights of people. I can believe that this is only roughly true! Actually, the usual way we come to a normal distribution is from the central limit theorem (CLT); in practice we want something like the mechanism of the CLT to apply. When do we get the CLT? Sure: If for some positive integer n our random variable Y is the sum, divided by the square root of n, of n independent, identically distributed (i.i.d.) samples of some distribution with, say, a mean and a finite variance. If n is 12, then can start to entertain normality if don't want accuracy in the tails. If want high accuracy in the tails of the normal distribution from the CLT, then I'd recommend some careful work and otherwise not trust the accuracy. A place where have a better shot at getting accuracy in a tail is the exponential distribution. The leading case is: Suppose we have a Geiger counter that goes "click" when it detects a radioactive decay. If the click rate is low so that the chances of two or more decays giving only one click are low, and real random variable T is the time until the next click, then under usual situations in practice T will have quite accurately exponential distribution. More generally, the stochastic process of such clicks is a Poisson process and an example of an arrival process. Well with weak assumptions, for positive integer n, the sum of n independent arrival processes approaches a Poisson process as n approaches infinity. This result is the renewal theorem with a proof in W. Feller's volume II. An example of a use of the renewal theorem is arrivals at a Web site. So, for each person on the planet there is an arrival process for that person at that site. If assume that the people are independent, then the Web site sees the sum of those arrivals and should see, over intervals of time much shorter than one day, a good approximation to a Poisson process. So, that's some 'applied probability' where we might work with tails. Mostly in applied probability, just f'get about accuracy in the tails!