aarjaneiro/planetmath huggingface dataset:
proof that transition functions of cotangent bundle are valid

proof that transition functions of cotangent bundle are valid

In this entry, we shall verify that the

transition functions

proposed for the

cotangent bundle

the three criteria required by the classical definition of a

manifold

.

The first criterion is the easiest to verify. If

α

=

β

, then

σ

α

⁢

α

reduces to the

identity

and we have

(

σ

′

α

⁢

α

⁢

(

x

1

,

…

,

x

2

⁢

n

)

)

i

=

(

σ

α

⁢

α

⁢

(

x

1

,

…

,

x

n

)

)

i

=

x

i

1

≤

i

≤

n

(

σ

′

α

⁢

α

⁢

(

x

1

,

…

,

x

2

⁢

n

)

)

i

+

n

=

∑

j

=

1

n

∂

(

σ

α

⁢

α

(

x

1

,

…

,

x

n

)

)

i

∂

⁡

x

j

⁢

x

j

+

n

=

∑

j

=

1

n

∂

⁡

x

i

∂

⁡

x

j

⁢

x

j

+

n

=

x

i

+

n

1

≤

i

≤

n

Thus we see that

σ

′

α

⁢

α

is the

identity map

, as required.

Next, we turn our attention to the third criterion — showing that

σ

′

β

⁢

γ

∘

σ

′

α

⁢

β

=

σ

′

α

⁢

γ

. For clarity of notation let us define

y

i

=

(

σ

′

α

⁢

β

)

i

⁢

(

x

1

,

…

⁢

x

2

⁢

n

)

. Then we have

(

σ

′

β

⁢

γ

∘

σ

′

α

⁢

β

)

i

⁢

(

x

1

,

…

,

x

2

⁢

n

)

=

(

σ

′

β

⁢

γ

)

i

⁢

(

y

1

,

…

,

y

2

⁢

n

)

=

(

σ

β

⁢

γ

)

i

⁢

(

y

1

,

…

,

y

n

)

=

(

σ

β

⁢

γ

∘

σ

α

⁢

β

)

i

⁢

(

x

1

,

…

,

x

n

)

=

(

σ

α

⁢

γ

)

i

⁢

(

x

1

,

…

,

x

n

)

=

(

σ

′

α

⁢

γ

)

i

⁢

(

x

1

,

…

,

x

2

⁢

n

)

when

1

≤

i

≤

n

.

(

σ

′

β

⁢

γ

∘

σ

′

α

⁢

β

)

i

+

n

⁢

(

x

1

,

…

,

x

2

⁢

n

)

=

(

σ

′

β

⁢

γ

)

i

+

n

⁢

(

y

1

,

…

,

y

2

⁢

n

)

=

∑

j

=

1

n

∂

(

σ

β

⁢

γ

(

y

1

,

…

,

y

n

)

)

i

∂

⁡

y

j

⁢

y

j

+

n

=

∑

j

=

1

n

∑

k

=

1

n

∂

(

σ

β

⁢

γ

(

y

1

,

…

,

y

n

)

)

i

∂

⁡

y

j

⁢

∂

(

σ

α

⁢

β

(

x

1

,

…

,

x

n

)

)

j

∂

⁡

x

k

⁢

x

n

+

k

=

∑

k

=

1

n

∂

(

σ

β

⁢

γ

∘

σ

α

⁢

β

(

x

1

,

…

,

x

n

)

)

i

∂

⁡

x

k

⁢

x

n

+

k

=

∑

k

=

1

n

∂

(

σ

α

⁢

γ

(

x

1

,

…

,

x

n

)

)

i

∂

⁡

x

k

⁢

x

n

+

k

=

σ

′

α

⁢

γ

⁢

(

x

1

,

…

,

x

2

⁢

n

)

when

1

≤

i

≤

n

.

Finally, the second criterion does not need to be checked because it is a

consequence

of the first and third criteria.

Title

proof that transition functions of cotangent bundle are valid

Canonical name

ProofThatTransitionFunctionsOfCotangentBundleAreValid

Date of creation

2013-03-22 14:52:25

Last modified on

2013-03-22 14:52:25

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

13

Author

rspuzio (6075)

Entry type

Proof

Classification

msc 58A32

Generated on Sat Feb 10 11:50:46 2018 by

LaTeXML
tangent bundle

tangent bundle

Let

M

be a

differentiable manifold

. Let the tangent bundle

T

⁢

M

of

M

be(as a set) the

disjoint union

∐

m

∈

M

T

m

⁢

M

of all the

tangent spaces

to

M

, i.e., the set of pairs

{

(

m

,

x

)

|

m

∈

M

,

x

∈

T

m

⁢

M

}

.

This naturally has a

manifold

structure

, given as follows. For

M

=

ℝ

n

,

T

⁢

ℝ

n

is obviously

isomorphic

to

ℝ

2

⁢

n

, and is thus obviously a manifold. By the

definition

of a differentiable manifold, for any

m

∈

M

, there is a neighborhood

U

of

m

and a

diffeomorphism

φ

:

ℝ

n

→

U

. Since this map is a diffeomorphism, its

derivative

is an

isomorphism

at all points. Thus

T

⁢

φ

:

T

⁢

ℝ

n

=

ℝ

2

⁢

n

→

T

⁢

U

is

bijective

, which endows

T

⁢

U

with a natural structure of a differentiable manifold. Since the transition maps for

M

are

differentiable

, they are for

T

⁢

M

as well, and

T

⁢

M

is a differentiable manifold. In fact, the projection

π

:

T

⁢

M

→

M

forgetting the

tangent vector

and remembering the point, is a vector bundle. A vector field on

M

is simply a

section

of this bundle.

The tangent bundle is functorial in the

obvious

sense: If

f

:

M

→

N

is differentiable, we get a map

T

⁢

f

:

T

⁢

M

→

T

⁢

N

, defined by

f

on the base, and its derivative on the fibers.

Title

tangent bundle

Canonical

name

TangentBundle

Date of creation

2013-03-22 13:58:59

Last modified on

2013-03-22 13:58:59

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

5

Author

bwebste (988)

Entry type

Definition

Classification

msc 58A32

Related topic

VectorField

Related topic

LieAlgebroids

Generated on Sat Feb 10 11:50:48 2018 by

LaTeXML
Finsler geometry

Finsler geometry

Let

ℳ

be an

n

-dimensional

differential manifold

and let

ϕ

:

T

⁢

ℳ

→

ℝ

be a function

ϕ

⁢

(

x

,

ξ

)

defined for

x

∈

ℳ

and

ξ

∈

T

x

⁢

ℳ

such that

ϕ

⁢

(

x

,

⋅

)

is a possibly non symmetric norm on

T

x

⁢

ℳ

. The couple

(

ℳ

,

ϕ

)

is called a Finsler space.

Let us define the

ϕ

-

length of curves

in

ℳ

. If

γ

:

[

a

,

b

]

→

ℳ

is a

differentiable

curve we define

ℓ

ϕ

⁢

(

γ

)

:=

∫

a

b

ϕ

⁢

(

γ

′

⁢

(

t

)

)

⁢

𝑑

t

.

So a natural geodesic distance can be defined on

ℳ

which turns the Finsler space into a

quasi-metric space

(if

ℳ

is

connected

):

d

ϕ

⁢

(

x

,

y

)

:=

inf

⁡

{

ℓ

ϕ

⁢

(

γ

)

:

γ

is a differentiable curve

γ

:

[

a

,

b

]

→

ℳ

such that

γ

⁢

(

a

)

=

x

and

γ

⁢

(

b

)

=

y

}

.

Notice that every

Riemann

manifold

(

ℳ

,

g

)

is also a Finsler space,
the norm

ϕ

⁢

(

x

,

⋅

)

being the norm

induced

by the scalar product

g

⁢

(

x

)

.

A finite dimensional

Banach space

is another simple example of Finsler space, where

ϕ

⁢

(

x

,

ξ

)

:=

∥

ξ

∥

. Wulff Theorem is one of the most important theorems in this

ambient space

.

Title

Finsler geometry

Canonical name

FinslerGeometry

Date of creation

2013-03-22 15:03:37

Last modified on

2013-03-22 15:03:37

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

8

Author

paolini (1187)

Entry type

Definition

Classification

msc 58B20

Classification

msc 53B40

Classification

msc 53C60

Related topic

WulffTheorem

Generated on Sat Feb 10 11:50:50 2018 by

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general Stokes theorem

general Stokes theorem

Let

M

be an

oriented

r

-dimensional

differentiable manifold

with
a

piecewise

differentiable

boundary

∂

⁡

M

. Further, let

∂

⁡

M

have the
orientation induced by

M

.
If

ω

is an

(

r

-

1

)

-form on

M

with compact support, whose components
have

continuous

first

partial derivatives

in any

coordinate chart

, then

∫

M

d

ω

=

∫

∂

⁡

M

ω

.

Title

general Stokes theorem

Canonical name

GeneralStokesTheorem

Date of creation

2013-03-22 12:44:52

Last modified on

2013-03-22 12:44:52

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

11

Author

matte (1858)

Entry type

Theorem

Classification

msc 58C35

Synonym

Stokes theorem

Related topic

DifferentialForms

Related topic

GaussGreenTheorem

Related topic

ClassicalStokesTheorem

Generated on Sat Feb 10 11:50:59 2018 by

LaTeXML
spectral radius

spectral radius

If

V

is a

vector space

over

ℂ

, the

spectrum

of a

linear mapping

T

:

V

→

V

is the set

σ

⁢

(

T

)

=

{

λ

∈

ℂ

:

T

-

λ

⁢

I

⁢

is not invertible

}

,

where

I

denotes the

identity mapping

.
If

V

is finite dimensional, the spectrum of

T

is precisely the set of its

eigenvalues

. For infinite dimensional spaces this is not generally true,
although it is true that each eigenvalue of

T

belongs to

σ

⁢

(

T

)

. The

spectral radius

of

T

is

ρ

⁢

(

T

)

=

sup

⁡

{

|

λ

|

:

λ

∈

σ

⁢

(

T

)

}

.

More generally, the spectrum and spectral radius can be defined for

Banach algebras

with

identity element

: If

𝒜

is a Banach algebra over

ℂ

with identity element

e

, the spectrum of an

element

a

∈

𝒜

is the set

σ

⁢

(

a

)

=

{

λ

∈

ℂ

:

a

-

λ

⁢

e

⁢

is not invertible in

⁢

𝒜

}

The spectral radius of

a

is

ρ

⁢

(

a

)

=

sup

⁡

{

|

λ

|

:

λ

∈

σ

⁢

(

a

)

}

.

Title

spectral radius

Canonical name

SpectralRadius

Date of creation

2013-03-22 13:13:58

Last modified on

2013-03-22 13:13:58

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

11

Author

Koro (127)

Entry type

Definition

Classification

msc 58C40

Defines

spectrum

Generated on Sat Feb 10 11:51:04 2018 by

LaTeXML
Morse complex

Morse complex

Let

M

be a

smooth manifold

, and

u

:

M

→

ℝ

be a Morse function. Let

C

n

u

⁢

(

M

)

be a

vector space

of formal

ℂ

-

linear combinations

of

critical points

of

u

with index

n

. Then there exists a differential

∂

n

:

C

n

→

C

n

-

1

based on the Morse flow making

C

*

into a chain complex called the

Morse complex

such that the

homology

of the complex is the

singular homology

of

M

. In particular, the number of critical points of

u

of index

n

on

M

is at least the

n

-th

Betti number

, and the

alternating sum

of the number of critical points of

u

is the

Euler characteristic

of

M

.

Title

Morse complex

Canonical name

MorseComplex

Date of creation

2013-03-22 13:53:18

Last modified on

2013-03-22 13:53:18

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

4

Author

bwebste (988)

Entry type

Definition

Classification

msc 58E05

Generated on Sat Feb 10 11:51:06 2018 by

LaTeXML
Morse function

Morse function

Let

M

be a

smooth manifold

. A

critical point

of a map

u

:

M

→

ℝ

at

x

∈

M

is called if the

Hessian matrix

H

u

(in any local coordinate system) at

x

is non-degenerate.

A

smooth function

u

:

M

→

ℝ

is called

Morse

if all its critical points are non-degenerate.

Morse functions

exist on any smooth manifold, and in fact form an open dense (

http://planetmath.org/Dense

) subset of smooth functions on

M

(this fact is often phrased “a generic smooth function is Morse”).

Title

Morse function

Canonical name

MorseFunction

Date of creation

2013-03-22 13:53:15

Last modified on

2013-03-22 13:53:15

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

8

Author

bwebste (988)

Entry type

Definition

Classification

msc 58E05

Defines

non-degenerate critical point

Generated on Sat Feb 10 11:51:08 2018 by

LaTeXML
Morse lemma

Morse lemma

Let

M

be a smooth

n

-dimensional manifold, and

f

:

M

→

ℝ

a

smooth map

. We denote by

Crit

⁢

(

f

)

the set of critical points of

f

, i.e.

Crit

⁢

(

f

)

=

{

p

∈

M

|

(

f

*

)

p

=

0

}

For each

p

∈

Crit

⁢

(

f

)

we denote by

f

*

*

:

T

p

⁢

M

×

T

p

⁢

M

→

ℝ

(or

(

f

*

*

)

p

if

p

need to be specified) the
bilinear map

f

*

*

⁢

(

v

,

w

)

=

v

⁢

(

w

~

⁢

(

f

)

)

=

w

⁢

(

v

~

⁢

(

f

)

)

,

∀

v

,

w

∈

T

p

⁢

M

,

where

v

~

,

w

~

∈

𝒯

⁢

(

M

)

are
smooth

vector fields

such that

v

~

p

=

v

and

w

~

p

=

w

.
This is a good

definition

. In fact

p

∈

Crit

⁢

(

f

)

implies

v

⁢

(

w

~

⁢

(

f

)

)

-

w

⁢

(

v

~

⁢

(

f

)

)

=

(

v

~

⁢

(

f

)

,

w

~

⁢

(

f

)

)

p

=

f

*

⁢

(

v

~

,

w

~

)

p

=

0

.

In smooth local coordinates

x

1

,

…

,

x

n

on a

neighborhood

U

of

p

we have

f

*

*

⁢

(

∂

∂

⁡

x

i

|

p

,

∂

∂

⁡

x

j

|

p

)

=

∂

2

⁡

f

∂

⁡

x

i

⁢

∂

⁡

x

j

⁢

(

p

)

.

A critical point

p

∈

Crit

⁢

(

f

)

is called

non degenerate

when the matrix

(

∂

2

⁡

f

∂

⁡

x

i

⁢

∂

⁡

x

j

⁢

(

p

)

)

i

,

j

∈

{

1

,

…

,

n

}

is non

singular

. We can equivalently
express this condition without the use of local coordinates saying
that

p

∈

Crit

⁢

(

f

)

is non degenerate when for each

v

∈

T

p

⁢

M

∖

{

0

}

the

linear functional

f

*

*

⁢

(

v

,

⋅

)

∈

Hom

⁢

(

T

p

⁢

M

,

ℝ

)

is not zero, i.e. there exists

w

such that

f

*

*

⁢

(

v

,

w

)

≠

0

.

We recall that the

index

of a bilinear functional

H

:

V

×

V

→

ℝ

is the

dimension

Index

⁢

(

H

)

of a maximal

linear
subspace

W

⊆

V

such that

H

is

negative definite

on

W

×

W

.

Theorem 1

(Morse lemma)

Let

f

:

M

→

R

be a smooth map. For each non degenerate

p

∈

Crit

⁢

(

f

)

there exists a neighborhood

U

of

p

and smooth

coordinates

x

=

(

x

1

,

…

,

x

n

)

on

U

such that

x

⁢

(

p

)

=

0

and

f

|

U

=

f

⁢

(

p

)

-

(

x

1

)

2

-

…

-

(

x

λ

)

2

+

(

x

λ

+

1

)

2

+

…

+

(

x

n

)

2

,

where

λ

=

Index

⁢

(

(

f

*

*

)

p

)

.

Title

Morse lemma

Canonical

name

MorseLemma

Date of creation

2013-03-22 13:53:12

Last modified on

2013-03-22 13:53:12

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

18

Author

matte (1858)

Entry type

Theorem

Classification

msc 58E05

Defines

non degenerate critical point

Defines

index of a bilinear map

Generated on Sat Feb 10 11:51:10 2018 by

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centralizer

centralizer

Let

G

a group acting on itself by

conjugation

.
Let

X

be a subset of

G

.
The

stabilizer

of

X

is called the

centralizer

of

X

and it’s the set

C

G

⁢

(

X

)

=

{

g

∈

G

:

g

⁢

x

⁢

g

-

1

=

x

for all

⁢

x

∈

X

}

For any group

G

,

C

G

⁢

(

G

)

=

Z

⁢

(

G

)

, the center of

G

. Thus, any

subgroup

of

C

G

⁢

(

G

)

is an

abelian

subgroup of

G

. However, the converse is generally not true. For example, take any

non-abelian group

and pick any element not in the center. Then the subgroup generated by it is obviously abelian, clearly non-trivial and not contained in the center.

Title

centralizer

Canonical name

Centralizer1

Date of creation

2013-03-22 14:01:20

Last modified on

2013-03-22 14:01:20

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

8

Author

yark (2760)

Entry type

Definition

Classification

msc 58E40

Related topic

CentralizersInAlgebra

Generated on Sat Feb 10 11:51:13 2018 by

LaTeXML
Bayes’ theorem

Bayes’ theorem

\PMlinkescapephrase

states

Let

(

A

n

)

be a sequence of

mutually exclusive events

whose union (

http://planetmath.org/Union

) is the

sample space

and let

E

be any event. All of the events have nonzero

probability

(

P

⁢

(

E

)

>

0

and

P

⁢

(

A

n

)

>

0

for all

n

). Bayes’ Theorem states

P

(

A

j

|

E

)

=

P

(

A

j

)

P

(

E

|

A

j

)

∑

i

P

(

A

i

)

P

(

E

|

A

i

)

for any

A

j

∈

(

A

n

)

.

A simpler formulation is:

P

(

A

|

B

)

=

P

(

B

|

A

)

P

(

A

)

P

⁢

(

B

)

For two events,

A

and

B

(also with nonzero probability).

References

1

Milton, J.S., Arnold, Jesse C.,

Introduction to Probability and Statistics: Principles and Applications for Engineering and the Computing Sciences

, McGraw Hill, 1995.

Title

Bayes’ theorem

Canonical name

BayesTheorem

Date of creation

2013-03-22 12:02:13

Last modified on

2013-03-22 12:02:13

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

11

Author

akrowne (2)

Entry type

Theorem

Classification

msc 60-00

Classification

msc 62A01

Synonym

Bayes’ Rule

Related topic

ConditionalProbability

Generated on Sat Feb 10 11:51:42 2018 by

LaTeXML
Bernoulli random variable

Bernoulli random variable

X

is a

Bernoulli random variable

with

parameter

p

if

f

X

⁢

(

x

)

=

p

x

⁢

(

1

-

p

)

1

-

x

,

x

=

{

0

,

1

}

Parameters:

⋆

p

∈

[

0

,

1

]

Syntax:

X

∼

B

⁢

e

⁢

r

⁢

n

⁢

o

⁢

u

⁢

l

⁢

l

⁢

i

⁢

(

p

)

Notes:

1.

X

represents the number of successful results in a Bernoulli trial. A Bernoulli trial is an experiment in which only two outcomes are possible: success, with

probability

p

, and failure, with probability

1

-

p

.

2.

E

⁢

[

X

]

=

p

3.

V

⁢

a

⁢

r

⁢

[

X

]

=

p

⁢

(

1

-

p

)

4.

M

X

⁢

(

t

)

=

p

⁢

e

t

+

(

1

-

p

)

Title

Bernoulli random variable

Canonical

name

BernoulliRandomVariable

Date of creation

2013-03-22 11:54:01

Last modified on

2013-03-22 11:54:01

Owner

Riemann (23)

Last modified by

Riemann (23)

Numerical id

7

Author

Riemann (23)

Entry type

Definition

Classification

msc 60-00

Synonym

Bernoulli distribution

Related topic

BernoulliDistribution2

Related topic

GeometricDistribution2

Generated on Sat Feb 10 11:51:44 2018 by

LaTeXML
beta random variable

beta random variable

X

is a

beta random variable

with

parameters

a and b

if

f

X

⁢

(

x

)

=

x

a

-

1

⁢

(

1

-

x

)

b

-

1

β

⁢

(

a

,

b

)

,

x

∈

[

0

,

1

]

Parameters:

⋆

a

>

0

⋆

b

>

0

Syntax:

X

∼

B

⁢

e

⁢

t

⁢

a

⁢

(

a

,

b

)

Notes:

1.

X

is used in many

statistical models

.

2.

The function

β

:

R

×

R

→

R

is defined as

β

⁢

(

a

,

b

)

=

∫

0

1

x

a

-

1

⁢

(

1

-

x

)

b

-

1

⁢

𝑑

x

.

β

⁢

(

a

,

b

)

can be calculated as

β

⁢

(

a

,

b

)

=

Γ

⁢

(

a

)

⁢

Γ

⁢

(

b

)

Γ

⁢

(

a

+

b

)

(For

information

on the

Γ

function, see the

gamma random variable

)

3.

E

⁢

[

X

]

=

a

a

+

b

4.

V

⁢

a

⁢

r

⁢

[

X

]

=

a

⁢

b

(

a

+

b

+

1

)

⁢

(

a

+

b

)

2

5.

M

X

⁢

(

t

)

not useful

Title

beta random variable

Canonical

name

BetaRandomVariable

Date of creation

2013-03-22 11:54:30

Last modified on

2013-03-22 11:54:30

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

11

Author

mathcam (2727)

Entry type

Definition

Classification

msc 60-00

Synonym

beta distribution

Generated on Sat Feb 10 11:51:46 2018 by

LaTeXML
bibliography for probability

bibliography for probability

References for Probability, MSC 60 - XX

Basics without Measure Theory, 60A, 60C

1.

Morris H. DeGroot, Mark J. Schervish

Probability

and

, 3rd ed., Addison Wesley, 2001

2.

Sheldon Ross,

A First Course in Probability

, 6th ed., Prentice Hall, New Jersey, 2002

3.

Harald CramÃÂ©r,

Sannolikhetskalkylen och några av dess användningar

, 2nd ed., Almqvist & Wicksell, Uppsala, 1951

4.

Harald CramÃÂ©r,

Mathematical Methods of

Statistics

, Princeton, 1946

Foundations, MSC 60A, 60E, 60F

1.

William Feller

,

An

Introduction

to Probability Theory and Its Applications,
Volume 1

, 3rd ed., John Wiley and Sons, Inc., New York, 1968.

2.

William Feller,

An Introduction to Probability Theory and Its Applications,
Volume 2

, 2rd ed., John Wiley and Sons, Inc., New York, 1971.

3.

Andrei N. Kolmogorov,

Foundations

of the Theory of Probability

,
2nd ed.,

American Mathematical Society

, 1997.

4.

Richard Durrett,

Probability: Theory and Examples

2nd
ed., Duxbury Press, 1995

5.

Kai Lai Chung,

A Course in Probability Theory

, 3rd ed. Academic Press, 2001.

6.

Harald CramÃÂ©r,

Random variables

and probability distributions

, Cambridge, 1937

Stochastic Processes, MSC 60G, 60H

1.

Joseph L. Doob,

Stochastic Processes

,
John Wiley and Sons, Inc. New York, 1953.

2.

Samuel. Karlin and Howard E. Taylor,

A First Course in Stochastic
Processes

, 2nd ed., Academic Press, New York, 1975.

3.

Samuel Karlin and Howard E. Taylor,

A Second Course in Stochastic Processes

,
Academic Press, Orlando, 1981.

4.

Paul G. Hoel, Sidney C. Port, Charles J. Stone,

Introduction to Stochastic
Processes

, Waveland Press, 1986.

5.

Frank Spitzer,

Principles of

Random Walk

, 2nd ed.,
Springer-Verlag, 2001

Applications, MSC 60J, 60K

1.

Sheldon M. Ross

Introduction to Probability Models

, 8th
ed., Academic Press, San Diego, 2003

2.

David Freedman,

Brownian Motion

and Diffusion

, Holden-Day, Inc.,
San Francisco, 1971

3.

Sheldon M. Ross

Simulation

, 3rd ed., Academic Press, San Diego, 2003

4.

David R. Cox

Renewal Theory

, Chapman & Hall, London, 1962

5.

Robert B. Cooper

Introduction to Queueing Theory

, Methuen, London,
1962; or check out the

http://www.cse.fau.edu/ bob/publications/IntroToQueueingTheory_Cooper.pdf

electronic
version

Title

bibliography for probability

Canonical name

BibliographyForProbability

Date of creation

2013-03-22 15:06:28

Last modified on

2013-03-22 15:06:28

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Bibliography

Classification

msc 60-00

Generated on Sat Feb 10 11:51:48 2018 by

LaTeXML
chi-squared random variable

chi-squared random variable

A

central

chi-squared random variable

X

with

n

>

0

degrees of freedom

is given by the

probability density function

f

X

⁢

(

x

)

=

(

1

2

)

n

2

Γ

⁢

(

n

2

)

⁢

x

n

2

-

1

⁢

e

-

1

2

⁢

x

for

x

>

0

, where

Γ

represents the

gamma function

.

The

parameter

n

is usually, but not always, an

integer

, in which case the

distribution

is that of the sum of the squares of a

sequence

of

n

independent

standard normal variables (

http://planetmath.org/NormalRandomVariable

)

X

1

,

X

2

,

…

,

X

n

,

X

=

X

1

2

+

X

2

2

+

⋯

+

X

n

2

.

Parameters:

n

∈

(

0

,

∞

)

.

Syntax:

X

∼

χ

(

n

)

2

Figure 1:

Densities

of the

chi-squared distribution

for different degrees of freedom.

Notes:

1.

This distribution is very widely used in

statistics

, such as in hypothesis tests and confidence intervals.

2.

The chi-squared distribution with

n

degrees of freedom is a result of evaluating the

gamma distribution

with

α

=

n

2

and

λ

=

1

2

.

3.

E

⁢

[

X

]

=

n

4.

Var

⁡

[

X

]

=

2

⁢

n

5.

The

moment generating function

is

M

X

⁢

(

t

)

=

(

1

-

2

⁢

t

)

-

n

2

,

and is defined for all

t

∈

ℂ

with

real part

(

http://planetmath.org/Complex

) less than

1

/

2

.

6.

The sum of independent

χ

(

m

)

2

and

χ

(

n

)

2

random variables

has the

χ

(

m

+

n

)

2

distribution.

Title

chi-squared random variable

Canonical

name

ChisquaredRandomVariable

Date of creation

2013-03-22 11:54:49

Last modified on

2013-03-22 11:54:49

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

16

Author

mathcam (2727)

Entry type

Definition

Classification

msc 60-00

Classification

msc 11-00

Classification

msc 20-01

Classification

msc 20A05

Synonym

central chi-squared distribution

Related topic

ChiSquaredStatistic

Generated on Sat Feb 10 11:51:50 2018 by

LaTeXML
continuous density function

continuous density function

Let

X

be a

continuous random variable

. The function

f

X

:

ℝ

→

[

0

,

1

]

defined as

f

X

⁢

(

x

)

=

∂

⁡

F

X

∂

⁡

x

, where

F

X

⁢

(

x

)

is the

cumulative distribution function

(

http://planetmath.org/CumulativeDistributionFunction

) of

X

, is called the

continuous density function

of

X

.
Please note that if

X

is a continuous random variable, then

f

X

⁢

(

x

)

does not equal

P

[

X

=

x

]

; for more information read the article on cumulative distribution functions.

Analogously to the discrete case, this function must satisfy:

1.

f

X

⁢

(

x

)

≥

0

for all

x

2.

∫

x

f

X

⁢

(

x

)

⁢

𝑑

x

=

1

Title

continuous density function

Canonical name

ContinuousDensityFunction

Date of creation

2013-03-22 11:53:16

Last modified on

2013-03-22 11:53:16

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Definition

Classification

msc 60-00

Classification

msc 81-00

Classification

msc 18-00

Synonym

mass function

Generated on Sat Feb 10 11:51:55 2018 by

LaTeXML