type of a distribution function

type of a distribution function

Two

distribution functions

F

,

G

:

ℝ

→

[

0

,

1

]

are said to of the same

type

if there exist

a

,

b

∈

ℝ

such that

G

⁢

(

x

)

=

F

⁢

(

a

⁢

x

+

b

)

.

a

is called the

scale parameter

, and

b

the

location parameter

or

centering

parameter

. Let’s write

F

=

t

G

to denote that

F

and

G

are of the same type.

Remarks

.

•

Necessarily

a

>

0

, for otherwise at least one of

G

⁢

(

-

∞

)

=

0

or

G

⁢

(

∞

)

=

1

would be violated.

•

If

G

⁢

(

x

)

=

F

⁢

(

x

+

b

)

, then the graph of

G

is

shifted

to the right from the graph of

F

by

b

units, if

b

>

0

and to the left if

b

<

0

.

•

If

G

⁢

(

x

)

=

F

⁢

(

a

⁢

x

)

, then the graph of

G

is

stretched

from the graph of

F

by

a

units if

a

>

1

, and

compressed

if

a

<

1

.

•

If

X

and

Y

are

random variables

whose distribution functions are of the same type, say,

F

and

G

respectively, and related by

G

⁢

(

x

)

=

F

⁢

(

a

⁢

x

+

b

)

, then

X

and

a

⁢

Y

+

b

are

identically distributed

, since

P

(

X

≤

z

)

=

F

(

z

)

=

G

(

z

-

b

a

)

=

P

(

Y

≤

z

-

b

a

)

=

P

(

a

Y

+

b

≤

z

)

.

When

X

and

a

⁢

Y

+

b

are identically distributed, we write

X

=

t

Y

.

•

Again, suppose

X

and

Y

correspond to

F

and

G

, two distribution functions of the same type related by

G

⁢

(

x

)

=

F

⁢

(

a

⁢

x

+

b

)

. Then it is easy to see that

E

⁢

[

X

]

<

∞

iff

E

⁢

[

Y

]

<

∞

. In fact, if the

expectation

exists for one, then

E

⁢

[

X

]

=

a

⁢

E

⁢

[

Y

]

+

b

. Furthermore,

V

⁢

a

⁢

r

⁢

[

X

]

is finite iff

V

⁢

a

⁢

r

⁢

[

Y

]

is. And in this case,

V

⁢

a

⁢

r

⁢

[

X

]

=

a

2

⁢

V

⁢

a

⁢

r

⁢

[

Y

]

. In general, convergence of moments is a “typical”

property

.

•

We can

partition

the set of distribution functions into disjoint subsets of

functions

belonging to the same types, since the

binary relation

=

t

is an

equivalence relation

.

•

By the same token, we can classify all real random variables defined on a fixed

probability space

according to their distribution functions, so that if

X

and

Y

are of the same type

τ

iff their corresponding distribution functions

F

and

G

are of type

τ

.

•

Given an

equivalence class

of distribution functions belonging to a certain type

τ

, such that a random variable

Y

of type

τ

exists with finite expectation and

variance

, then there is one distribution function

F

of type

τ

corresponding to a random variable

X

such that

E

⁢

[

X

]

=

0

and

V

⁢

a

⁢

r

⁢

[

X

]

=

1

.

F

is called the

standard distribution function

for type

τ

. For example, the standard (cumulative)

normal distribution

is the standard distribution function for the type consisting of all

normal distribution functions

.

•

Within each type

τ

, we can further classify the distribution functions: if

G

⁢

(

x

)

=

F

⁢

(

x

+

b

)

, then we say that

G

and

F

belong to the same

location family

under

τ

; and if

G

⁢

(

x

)

=

F

⁢

(

a

⁢

x

)

, then we say that

G

and

F

belong to the same

scale family

(under

τ

).

Title

type of a distribution function

Canonical

name

TypeOfADistributionFunction

Date of creation

2013-03-22 16:25:48

Last modified on

2013-03-22 16:25:48

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Definition

Classification

msc 60E05

Classification

msc 62E10

Synonym

centering factor

Synonym

scale parameter

Synonym

location parameter

Defines

type

Defines

scale factor

Defines

location factor

Defines

standard distribution function

Defines

location family

Defines

scale family

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