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# NORMAL DISTRIBUTION

Module by: Ewa Paszek. E-mail the author

Summary: This course is a short series of lectures on Introductory Statistics. Topics covered are listed in the Table of Contents. The notes were prepared by Ewa Paszek and Marek Kimmel. The development of this course has been supported by NSF 0203396 grant.

## NORMAL DISTRIBUTION

The normal distribution is perhaps the most important distribution in statistical applications since many measurements have (approximate) normal distributions. One explanation of this fact is the role of the normal distribution in the Central Theorem.

Definition 1:
1. The random variable X has a normal distribution if its p.d.f. is defined by
f( x )= 1 σ 2π exp[ ( xμ ) 2 2 σ 2 ],<x<, f( x )= 1 σ 2π exp[ ( xμ ) 2 2 σ 2 ],<x<,
(1)
where μ μ and σ 2 σ 2 are parameters satisfying <μ<,0<σ< <μ<,0<σ< , and also where exp[ v ] exp[ v ] means e v e v .
2. Briefly, we say that X is N( μ, σ 2 ) N( μ, σ 2 )

### Proof of the p.d.f. properties

Clearly, f( x )>0 f( x )>0 . Let now evaluate the integral: I= 1 σ 2π exp[ ( xμ ) 2 2 σ 2 ] dx, I= 1 σ 2π exp[ ( xμ ) 2 2 σ 2 ] dx, showing that it is equal to 1. In the integral, change the variables of integration by letting z=( xμ )/σ z=( xμ )/σ . Then,

I= 1 2π e z 2 /2 dz, I= 1 2π e z 2 /2 dz, since I>0 I>0 , if I 2 =1 I 2 =1 , then I=1 I=1 .

Now I 2 = 1 2π [ e x 2 /2 dx ][ e y 2 /2 dy ], I 2 = 1 2π [ e x 2 /2 dx ][ e y 2 /2 dy ], or equivalently,

I 2 = 1 2π exp( x 2 + y 2 2 )dxdy . I 2 = 1 2π exp( x 2 + y 2 2 )dxdy .

Letting x=rcosθ,y=rsinθ x=rcosθ,y=rsinθ (i.e., using polar coordinates), we have

I 2 = 1 2π 0 2π 0 e r 2 /2 rdrdθ= 1 2π 0 2π dθ= 1 2π 2π=1. I 2 = 1 2π 0 2π 0 e r 2 /2 rdrdθ= 1 2π 0 2π dθ= 1 2π 2π=1.

The mean and the variance of the normal distribution is as follows:

E( X )=μ E( X )=μ and Var( X )= μ 2 + σ 2 μ 2 = σ 2 . Var( X )= μ 2 + σ 2 μ 2 = σ 2 .

That is, the parameters μ μ and σ 2 σ 2 in the p.d.f. are the mean and the variance of X.

#### Example 1

If the p.d.f. of X is

f( x )= 1 32π exp[ ( x+7 ) 2 32 ],<x<, f( x )= 1 32π exp[ ( x+7 ) 2 32 ],<x<, then X is N( 7,16 ) N( 7,16 )

That is, X has a normal distribution with a mean μ μ =-7, variance σ 2 σ 2 =16, and the moment generating function

M( t )=exp( 7t+8 t 2 ). M( t )=exp( 7t+8 t 2 ).

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#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

#### Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks