Common Continous DistibutionsCommon Continous Distibutions

Normal DistributionNormal Distribution

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Uniform

Exponential

Normal

The Normal distribution

We have already seen the normal distribution as a "bellshaped" distribution, but we can formalize this.

The normal or Gaussian distribution is acontinuous probability distribution with probabilitydensity function (pdf)

for .

We then show that by

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Background

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CommonDists

Uniform

Exponential

Normal

The Normal distribution

A normal random variable is (often) a finite average ofmany repeated, independent, identical trials.

Mean width of the next 50 hexamine pallets

Mean height of 30 students

Rotal yield of the next 10 runs of a chemicalprocess

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Background

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CommonDists

Uniform

Exponential

Normal

Normal Distribution's Center and Shape

Regardless of the values of and , the normal pdf hasthe following shape:

In other words, the distribution is centered around and

has an inflection point at .

In this way, the value of determines the center of ourdistribution and the value of deterimes the spread.

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Uniform

Exponential

Normal

Normal Distribution's Center and Shape

Here we can see what differences in and do to theshape of the shape of distribution

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Mean dna VarianceMean dna Variance

ofof

Normal DistributionNormal Distribution

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Uniform

Exponential

Normal

The Normal distribution

It is not obvious, but

 

 

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One poine before we go onOne poine before we go on

StandardizationStandardization

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Uniform

Exponential

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De nition

Standardization is the process of transforming a randomvariable, , into the signed number of standarddeviations by which it is is above its mean value.

has mean

 

 

 

has variance (and standard deviation)

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Normal

 

 

The Calculus I methods of evaluating integrals via anti-differentiation will fail when it comes to normal densities.They do not have anti-derivatives that are expressible interms of elementary functions.

This means we cannot find probabilities of aNormally distributed random variable by hand.

So, what is the solution?

Use computers or tables of values.

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Uniform

Exponential

Normal

 

 

The use of tables for evaluating normal probabilitiesdepends on the following relationship. If

,

where .

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Uniform

Exponential

Normal

Std. Normal

Standard Normal Distribution

The parameters are important in determining theprobability, but because the pdf of a normal randomvariable is difficult to work with we often use thedistribution with and as a reference point.

Definition: Standard Normal Distribution The standard normal distribution is a normaldistribution with and . It has pdf

`

We say that a random variable is a "standard normalrandom variable" if it follows a standard normaldistribution or that .

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Uniform

Exponential

Normal

Std. Normal

Standard Normal Distribution (cont)

It's worth pointing out the reason why the standardnormal distribution is important. There is no "closedform" for the cdf of a normal distribution.

In other words, since we can't finish this step:

we have to estimate the value each time. However, wehave already done this for standard normal randomvariables already in Table B.3

So if then .

The good news is that we can connect any normalprobabilities to the values we have for the standardnormal probabilities.

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Uniform

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Normal

Std. Normal

Standard Normal Distribution (cont)

These facts drive the connection between different normalrandom variables:

Key Facts: Converting Normal Distributions

If and then

  If and then

We use this connection as a way to avoid working with thenormal pdf directly.

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StandardNormal

Standard Normal Distribution (cont)

A rule of thumb in dealing with questions about findingprobabilities of Normally distributed probabilities of

:

(1) Translate that question to standard Normaldistribution. i.e.

(2) Look it up in a table

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Uniform

Exponential

Normal

Std. Normal

CDF of Standard Normal Distribution

The standard Normal distribution $ Z\sim N(0,1)$ plays animportant rule in finding probabilities associated with aNormal random variable. The CDF of a standard Normaldistribution is

Therefore, we can find probabilities for all normaldistributions by tabulating probabilities for only thestandard normal distribution. We will use a table of thestandard normal cumulative probability function.

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Uniform

Exponential

Normal

Std. Normal

Standard Normal Distribution (cont)

Example: Normal to Standard Normal

If then:

where the valeu 0.9332 if found from Table B.3

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Uniform

Exponential

Normal

Std. Normal

Standard Normal Distribution (cont)

Example: Standard normal probabilities

 

 

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Std. Normal53 / 58

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Std. Normal54 / 58

Background

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Std. Normal

Some useful tips about standard Normaldistribution

By symmetry of the standard Normaldistribution around zero

 

 

 

We can also do it reverse, find such that

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