Examples of Continuous Probability Distributions

Examples of continuous

probability distributions:

The normal and standard normal

The Normal Distribution

f(X)

Changing ¦Ì shifts the

distribution left or right.

Changing ¦Ò increases

or decreases the

spread.

X

The Normal Distribution:

as mathematical function (pdf)

f ( x) =

1

¦Ò 2¦Ð

Note constants:

¦Ð=3.14159

e=2.71828

1 x?? 2

? (

)

¦Ò

2

?e

This is a bell shaped

curve with different

centers and spreads

depending on ? and ¦Ò

The Normal PDF

It¡¯s a probability function, so no matter what the

values of ? and ¦Ò, must integrate to 1!

+¡Þ

¡Ò¦Ò

?¡Þ

1

2¦Ð

1 x?? 2

? (

)

? e 2 ¦Ò dx

=1

Normal distribution is defined by its mean

and standard dev.

+¡Þ

E(X)=? =

¡Òx¦Ò

?¡Þ

Var(X)=¦Ò2 =

+¡Þ

(

¡Ò

x2

?¡Þ

Standard Deviation(X)=¦Ò

1

2¦Ð

1 x?? 2

? (

)

2

¦Ò

?e

dx

1

¦Ò

2¦Ð

?e

?

1 x?? 2

(

)

2 ¦Ò

dx ) ? ?

2

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