# Why is the normal distribution considered unique?

• June 9th 2011, 08:11 AM
rainer
Why is the normal distribution considered unique?
As I have been told, the normal distribution is special because of 4 criteria:

1) The errors do not depend on the coordinate system.
2) Errors in perpendicular directions are independent.
3) Large errors are less likely than small errors.
4) The sum of the distribution the (CDF) over the infinite domain equals 1.

My question is: Aren't there a lot of bell-shaped curves out there that meet these criteria?

For example $\frac{1}{\pi(1+x^2)}$ meets the three first assumptions plus

$\int_{-\infty}^{\infty} \frac{1}{\pi(1+x^2)}=1$

This one is also a lot more tractable. So why is the normal distribution considered so unique and special?
• June 9th 2011, 08:53 AM
HallsofIvy
The real importance- and "uniquness"- of the Normal distribution is this: if we have a large number, n, of trials from any probability distribution (with finite moments) the probability distribution for the mean of the trials is approximately Normal with the same mean and with standard deviation equal to the standard deviation of the original distribution over the square root of n. That is, if you take a large enough sample the Normal distribution will work no matter what the "underlying" distribution is.
• June 9th 2011, 05:33 PM
rainer
That's the CLT you're referring to if I'm not mistaken.

You say "approximately normal." Could you just as well have said "approximately bell-shaped"? Or is there something special about 1/exp(-1/2 etc.) that makes the CLT true?