Difference between a long tail and normal distribution Wikipedia article about "long tail" says that:

A probability distribution is said to have a long tail, if a larger
  share of population rests within its tail than would under a normal
  distribution.

I am confused about this. Isn't a normal distribution exponential one?
Meaning even large values of x will have some corresponding value of y, thus making those values of x "rest within its tail"?
Or does this sentence mean that area of tail in long-tail-distribution is more than are of tails in a normal-distribution?
Also what would be the difference between a long-tailed and a fat-tailed distribution?
 A: Recall that the density function of the (standard) normal has shape $ke^{-x^2/2}$, where $k$ is a constant. If $x$ is at all large, $e^{-x^2/2}$ is very close to $0$. Compare this with even an exponential distribution, which (for the same variance) has density function $e^{-x}$ (for $x\ge 0$). Although $e^{-x}$ decays rapidly, it does so far more slowly than $e^{-x^2/2}$. 
And consider now the Cauchy distribution, which has density function of shape $\frac{k}{1+x^2}$. This decays far more slowly than $e^{-x}$, and very much more slowly than $e^{-x^2/2}$: it is very much long-tailed. Or consider the distribution with density $\frac{1}{(1+x)^2}$, for $x\ge 0$. This density function decays so slowly that the mean of a random variable with this distribution does not exist (informally, is infinite): the tail wags the dog.
Remark: If you look through the long list of named probability distributions in Wikipedia, you will find that many of them have density functions that in the long run decay far more slowly than the density function of the normal. 
