2

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?

radha
  • 51

1 Answers1

2

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.

André Nicolas
  • 507,029
  • 2
    Are you prepared for the fact that in all likelihood, tomorrow you dethrone Arturo from his throne of reputation? – Asaf Karagila May 05 '13 at 22:10
  • Thanks André. That was a nice answer. From your answer, the type of tail is decided by how fast the function decays. So essentially, its still counting the area of the tail. Am I right? – radha May 05 '13 at 23:50