# Can anyone answer these 2 Questions?

1. Nov 22, 2005

### jokerwild

Normal probability distribution has been used in "wide variety of practical applications" and "widely used in statistical inference" My question is two-fold:
(a) Why is the Normal distribution so important, particularly with respect to problems dealing with statistical inference?

(b) why is Normal distribution so useful?

2. Nov 22, 2005

### Hurkyl

Staff Emeritus
The central limit theorem.

3. Nov 22, 2005

### HallsofIvy

Staff Emeritus
As Hurkyl said "the Central limit theorem".
Whenever we apply mathematics to a "real life" problem, we have to establish a "mathematical model". Often, since "real life" problems involve measurement which is never exact (like mathematics is), there are several different models we could use and we have to decide which to use. To solve statistics problems, we have to decide on an underlying probability distribution. Sometimes there are clear reasons for using a binomial distribution or Poisson's distribution.
However, for the vast majority of statistics problems, the normal distribution will give the best fit precisely because of the "Central Limit Theorem"- one of very few mathematical theorems that say "this is a good model for certain types of applications".
The Central Limit Theorem says: "If you take a n samples from a given probability distribution (any distribution as long as it has finite mean,$x\overline$, and standard deviation, $\sigma$), the sum of those samples will be approximately normally distributed with mean $n x\overline$ and standard deviation $\sqrt{n}\sigma$ while the mean of the samples will be approximately normally distributed with mean $x\overline$ and standard deviation $\frac{\sigma}{\sqrt{n}}$.
The larger n is, the better the approximation.

Last edited: Nov 22, 2005
4. Nov 22, 2005

### jokerwild

Thank you very much.