aarjaneiro/planetmath huggingface dataset:
regular conditional probability

regular conditional probability

Introduction

Suppose

(

Ω

,

ℱ

,

P

)

is a

probability space

and

B

∈

ℱ

be an event with

P

⁢

(

B

)

>

0

. It is easy to see that

P

B

:

ℱ

→

[

0

,

1

]

defined by

P

B

(

A

)

:=

P

(

A

|

B

)

,

the

conditional probability

of event

A

given

B

, is a

probability measure

defined on

ℱ

, since:

1.

P

B

is clearly non-negative;

2.

P

B

⁢

(

Ω

)

=

P

⁢

(

Ω

∩

B

)

P

⁢

(

B

)

=

P

⁢

(

B

)

P

⁢

(

B

)

=

1

;

3.

P

B

is

countably additive

: for if

{

A

1

,

A

2

,

…

}

is a

countable

collection

of

pairwise disjoint

events in

ℱ

, then

P

B

⁢

(

⋃

i

=

1

∞

A

i

)

=

P

⁢

(

B

∩

(

⋃

A

i

)

)

P

⁢

(

B

)

=

P

⁢

(

⋃

(

B

∩

A

i

)

)

P

⁢

(

B

)

=

∑

P

⁢

(

B

∩

A

i

)

P

⁢

(

B

)

=

∑

i

=

1

∞

P

B

⁢

(

A

i

)

,

as

{

B

∩

A

1

,

B

∩

A

2

,

…

}

is a collection of pairwise disjoint events also.

Regular Conditional Probability

Can we extend the

definition

above to

P

𝒢

, where

𝒢

is a sub

sigma algebra

of

ℱ

instead of an event? First, we need to be careful what we mean by

P

𝒢

, since, given any event

A

∈

ℱ

,

P

(

A

|

𝒢

)

is not a real number. And

strictly

speaking, it is not even a

random variable

, but an

equivalence class

of random variables (each pair differing by a null event in

𝒢

).

With this in mind, we start with a probability measure

P

defined on

ℱ

and a sub sigma algebra

𝒢

of

ℱ

. A

function

P

𝒢

:

𝒢

×

Ω

→

[

0

,

1

]

is a called a

regular conditional probability

if it has the following

properties

:

1.

for each event

A

∈

𝒢

,

P

𝒢

⁢

(

A

,

⋅

)

:

Ω

→

[

0

,

1

]

is a conditional probability (

http://planetmath.org/ProbabilityConditioningOnASigmaAlgebra

) (as a random variable) of

A

given

𝒢

; that is,

(a)

P

𝒢

⁢

(

A

,

⋅

)

is

𝒢

-measurable (

http://planetmath.org/MathcalFMeasurableFunction

) and

(b)

for every

B

∈

𝒢

, we have

∫

B

P

𝒢

⁢

(

A

,

⋅

)

⁢

𝑑

P

=

P

⁢

(

A

∩

B

)

.

2.

for every outcome

ω

∈

Ω

,

P

𝒢

⁢

(

⋅

,

ω

)

:

𝒢

→

[

0

,

1

]

is a probability measure.

There are probability spaces where no regular conditional probabilities can be defined. However, when a regular conditional probability function does exist on a space

Ω

, then by condition 2 of the definition, we can define a “

conditional

” probability measure on

Ω

for each outcome in the sense of the first two paragraphs.

Title

regular conditional probability

Canonical

name

RegularConditionalProbability

Date of creation

2013-03-22 16:25:24

Last modified on

2013-03-22 16:25:24

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 60A99

Generated on Sat Feb 10 11:54:23 2018 by

LaTeXML
reverse Markov inequality

reverse Markov inequality

Let

X

be a

random variable

that

satisfies

Pr

⁡

(

X

≤

a

)

=

1

for some

constant

a

.
Then, for

d

<

E

⁢

[

X

]

,

Pr

⁡

(

X

>

d

)

≥

E

⁢

[

X

]

-

d

a

-

d

Proof:

Apply the Markov’s inequality to the random variable

X

~

=

a

-

X

,

Pr

⁡

(

X

≤

d

)

=

Pr

⁡

(

X

~

≥

a

-

d

)

≤

E

⁢

[

X

~

]

a

-

d

=

a

-

E

⁢

[

X

]

a

-

d

.

Hence

Pr

⁡

(

X

>

d

)

≥

1

-

a

-

E

⁢

[

X

]

a

-

d

=

E

⁢

[

X

]

-

d

a

-

d

.

Title

reverse Markov inequality

Canonical name

ReverseMarkovInequality

Date of creation

2013-03-22 17:48:08

Last modified on

2013-03-22 17:48:08

Owner

kshum (5987)

Last modified by

kshum (5987)

Numerical id

6

Author

kshum (5987)

Entry type

Definition

Classification

msc 60A99

Related topic

MarkovsInequality

Generated on Sat Feb 10 11:54:24 2018 by

LaTeXML
support

support

If

f

is a probability distribution on

Ω

then the

support

of

f

is

{

x

∈

Ω

⁢

∣

f

⁢

(

x

)

>

⁢

0

}

. That is, the set of

x

which have a

positive

probability

under the

distribution

.

Title

support

Canonical

name

Support1

Date of creation

2013-03-22 13:02:40

Last modified on

2013-03-22 13:02:40

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

6

Author

Henry (455)

Entry type

Definition

Classification

msc 60A99

Classification

msc 68Q30

Generated on Sat Feb 10 11:54:26 2018 by

LaTeXML
σ-algebra generated by a random variable

σ

-algebra generated by a random variable

Given the

probability space

(

Ω

,

ℱ

,

P

)

, any

random variable

X

:

Ω

→

ℝ

is

F

-

measurable (

http://planetmath.org/MeasurableFunctions

),  in the following sense:

X

-

1

⁢

(

U

)

=

{

ω

∈

Ω

:

X

⁢

(

ω

)

∈

U

}

∈

ℱ

for any open sets

U

⊆

ℝ

, or equivalently any

Borel sets

U

⊂

ℝ

.

We now define

ℱ

X

as follows:

ℱ

X

=

X

-

1

⁢

(

ℬ

)

:=

{

X

-

1

⁢

(

B

)

:

B

∈

ℬ

}

,

where

ℬ

is the Borel

σ

-

algebra

on

ℝ

.

ℱ

X

is sometimes denoted as

σ

⁢

(

X

)

.

ℱ

X

is a

sigma algebra

since it

satisfies

the following:

•

∅

=

X

-

1

⁢

(

∅

)

∈

ℱ

X

,

•

Ω

-

X

-

1

⁢

(

B

)

=

X

-

1

⁢

(

ℝ

-

B

)

∈

ℱ

X

, and

•

⋃

X

-

1

⁢

(

B

i

)

=

X

-

1

⁢

(

⋃

B

i

)

∈

ℱ

X

.

It is also clear that

ℱ

X

is the smallest

σ

-algebra containing all sets of the form

X

-

1

⁢

(

B

)

,

B

∈

ℬ

.

ℱ

X

as defined above is called the

σ

-algebra

X

.

Title

σ

-algebra generated by a random variable

Canonical name

sigmaalgebraGeneratedByARandomVariable

Date of creation

2013-03-22 15:48:19

Last modified on

2013-03-22 15:48:19

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

19

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 60A99

Classification

msc 60A10

Related topic

SigmaAlgebra

Generated on Sat Feb 10 11:54:28 2018 by

LaTeXML
convergence in probability

convergence in probability

Let

{

X

i

}

be a

sequence

of

random variables

defined on a

probability space

(

Ω

,

ℱ

,

P

)

taking values in a

separable

metric
space

(

Y

,

d

)

, where

d

is the metric. Then we say the sequence

X

i

converges in probability

or

converges in measure

to a random variable

X

if
for every

ε

>

0

,

lim

i

→

∞

P

(

d

(

X

i

,

X

)

≥

ε

)

=

0

.

We denote convergence in probability of

X

i

to

X

by

X

i

⟶

p

⁢

r

X

.

Equivalently,

X

i

⟶

p

⁢

r

X

iff every

subsequence

of

{

X

i

}

contains a subsequence which

converges

to

X

almost surely.

Remarks

.

•

Unlike ordinary convergence,

X

i

⟶

p

⁢

r

X

and

X

i

⟶

p

⁢

r

Y

only implies that

X

=

Y

almost surely.

•

The need for separability on

Y

is to ensure that the metric,

d

⁢

(

X

i

,

X

)

, is a random variable, for all random variables

X

i

and

X

.

•

Convergence almost surely implies convergence in probability but not

conversely

.

References

1

R. M. Dudley,

Real Analysis and

Probability

, Cambridge University Press (2002).

2

W. Feller,

An

Introduction

to Probability Theory and Its Applications. Vol. 1

, Wiley, 3rd ed. (1968).

Title

convergence in probability

Canonical

name

ConvergenceInProbability

Date of creation

2013-03-22 15:01:05

Last modified on

2013-03-22 15:01:05

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 60B10

Synonym

converge in probability

Synonym

converges in measure

Synonym

converge in measure

Synonym

convergence in measure

Defines

converges in probability

Generated on Sat Feb 10 11:54:30 2018 by

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multidimensional Gaussian integral

multidimensional Gaussian integral

Let

𝐱

=

[

x

1

⁢

x

2

⁢

…

⁢

x

n

]

T

and

d

n

⁢

𝐱

≡

∏

i

=

1

n

d

⁢

x

i

.

Theorem 1

Let

K

be a

symmetric

positive definite

(

http://planetmath.org/PositiveDefinite

) matrix and

f

:

R

n

→

R

, where

f

⁢

(

x

)

=

exp

⁡

(

-

1

2

⁢

x

T

⁢

K

-

1

⁢

x

)

.
Then

∫

e

-

1

2

⁢

𝐱

T

⁢

𝐊

-

1

⁢

𝐱

⁢

d

n

⁢

𝐱

=

(

(

2

⁢

π

)

n

⁢

|

𝐊

|

)

1

2

(1)

where

|

K

|

=

det

⁡

K

.

Proof.

𝐊

-

1

is real and symmetric (since

(

𝐊

-

1

)

T

=

(

𝐊

T

)

-

1

=

𝐊

-

1

)

. For convenience, let

𝐀

=

𝐊

-

1

. We can decompose

𝐀

into

𝐀

=

𝐓

⁢

𝚲

⁢

𝐓

-

1

, where

𝐓

is an orthonormal (

𝐓

T

⁢

𝐓

=

𝐈

) matrix of the

eigenvectors

of

𝐀

and

𝚲

is a

diagonal matrix

of the

eigenvalues

of

𝐀

. Then

∫

e

-

1

2

⁢

𝐱

T

⁢

𝐀𝐱

⁢

d

n

⁢

𝐱

=

∫

e

-

1

2

⁢

𝐱

T

⁢

𝐓

⁢

𝚲

⁢

𝐓

-

1

⁢

𝐱

⁢

d

n

⁢

𝐱

.

(2)

Because

𝐓

is orthonormal, we have

𝐓

-

1

=

𝐓

T

. Now define a new vector variable

𝐲

≡

𝐓

T

⁢

𝐱

, and substitute:

∫

e

-

1

2

⁢

𝐱

T

⁢

𝐓

⁢

𝚲

⁢

𝐓

-

1

⁢

𝐱

⁢

d

n

⁢

𝐱

=

∫

e

-

1

2

⁢

𝐱

T

⁢

𝐓

⁢

𝚲

⁢

𝐓

T

⁢

𝐱

⁢

d

n

⁢

𝐱

(3)

=

∫

e

-

1

2

⁢

𝐲

T

⁢

𝚲

⁢

𝐲

⁢

|

𝐉

|

⁢

d

n

⁢

𝐲

(4)

where

|

𝐉

|

is the

determinant

of the Jacobian matrix

J

m

⁢

n

=

∂

⁡

x

m

∂

⁡

y

n

. In this case,

𝐉

=

𝐓

and thus

|

𝐉

|

=

1

.

Since

𝚲

is

diagonal

, the integral may be separated into the product of

n

independent

Gaussian distributions

, each of which we can integrate separately using the well-known formula

∫

e

-

1

2

⁢

a

⁢

t

2

⁢

𝑑

t

=

(

2

⁢

π

a

)

1

2

.

(6)

Carrying out this program, we get

∫

e

-

1

2

⁢

𝐲

T

⁢

𝚲

⁢

𝐲

⁢

d

n

⁢

𝐲

=

∏

k

=

1

n

∫

e

-

1

2

⁢

λ

k

⁢

y

k

2

⁢

𝑑

y

k

(7)

=

∏

k

=

1

n

(

2

⁢

π

λ

k

)

1

2

(8)

=

(

(

2

⁢

π

)

n

∏

k

=

1

n

λ

k

)

1

2

(9)

=

(

(

2

⁢

π

)

n

|

𝚲

|

)

1

2

.

(10)

Now, we have

|

𝐀

|

=

|

𝐓

⁢

𝚲

⁢

𝐓

-

1

|

=

|

𝐓

|

⁢

|

𝚲

|

⁢

|

𝐓

-

1

|

=

|

𝚲

|

, so this becomes

∫

e

-

1

2

⁢

𝐱

T

⁢

𝐀𝐱

⁢

d

n

⁢

𝐱

=

(

(

2

⁢

π

)

n

|

𝐀

|

)

1

2

.

(12)

Substituting back in for

𝐊

-

1

, we get

∫

e

-

1

2

⁢

𝐱

T

⁢

𝐊

-

1

⁢

𝐱

⁢

d

n

⁢

𝐱

=

(

(

2

⁢

π

)

n

|

𝐊

-

1

|

)

1

2

=

(

(

2

⁢

π

)

n

⁢

|

𝐊

|

)

1

2

,

(13)

as promised.

Title

multidimensional Gaussian integral

Canonical name

MultidimensionalGaussianIntegral

Date of creation

2013-03-22 12:18:44

Last modified on

2013-03-22 12:18:44

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

22

Author

Mathprof (13753)

Entry type

Theorem

Classification

msc 60B11

Classification

msc 62H99

Classification

msc 62H10

Related topic

JacobiDeterminant

Related topic

AreaUnderGaussianCurve

Related topic

ProofOfGaussianMaximizesEntropyForGivenCovariance

Generated on Sat Feb 10 11:54:33 2018 by

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approximating the birthday problem

approximating the birthday problem

A straightforward

generalization

of the

birthday problem

can be stated as:

Given

N

possible outcomes, how many random samples must be taken so that with a

probability

of at least

1

/

2

there are two equal samples?

Answer: Approximately

1.2

⁢

N

samples must be taken.

So in the typical

birthday problem

setting the

N

=

365

– the number of days in the typical year, and the result is that 23 people in a group gives at least a 50% chance of two people with a common birthday. This result can be obtained by a concise computation of the probilities for various sizes of the group until the proper value of 23 is found, but this approach does not lend itself to finding the value of 23 efficiently.

The result however has an easy to compute approximation of

1.2

⁢

N

, so in the case of

N

=

365

, the result is approximated by

1.2

⁢

365

≈

23

. We now validate the estimate.

Recall the probability of selecting a duplicate from

N

possible outcomes within

k

choices is

p

⁢

(

N

,

k

)

=

(

N

k

)

⁢

1

N

k

=

(

1

-

1

N

)

⁢

(

1

-

2

N

)

⁢

⋯

⁢

(

1

-

k

-

1

N

)

.

As

e

x

=

lim

m

→

∞

⁡

(

1

+

x

m

)

m

we can roughly approximate the formal

product

above by:

p

⁢

(

N

,

k

)

≈

e

-

1

/

N

-

2

/

N

-

⋯

-

k

/

N

=

e

-

(

k

+

1

2

)

⁢

1

N

.

Now we are after a

formula

in which given

N

and the desired probability of

1

/

2

we can estimate

k

. So we solve for

k

in the equation:

1

2

=

p

⁢

(

N

,

k

)

=

e

-

(

k

2

)

⁢

1

N

=

e

-

k

⁢

(

k

-

1

)

2

⁢

N

;

-

2

⁢

N

⁢

ln

⁡

1

2

=

k

2

-

k

;

1

4

+

2

⁢

N

⁢

ln

⁡

2

=

k

2

-

k

+

1

4

=

(

k

-

1

2

)

2

;

k

=

1

2

+

1

4

+

2

⁢

N

⁢

ln

⁡

2

.

(1)

A less precise but easier to compute approximation comes from simplifying our original estimate further:

p

⁢

(

N

,

k

)

≈

e

-

k

⁢

(

k

-

1

)

2

⁢

N

≈

e

-

k

2

/

2

⁢

N

.

Then set

1

/

2

=

e

-

k

⁢

(

k

-

1

)

2

⁢

N

≈

e

-

k

2

/

2

⁢

N

and once
again solve for

k

to find:

k

=

2

⁢

N

⁢

ln

⁡

2

≈

1.2

⁢

N

.

We also see that if we wish to have an outcome with probability

ε

>

0

of no matches then we can estimate the sample size by:

ε

=

p

⁢

(

N

,

k

)

≈

e

-

k

2

/

2

⁢

N

;

k

≈

2

⁢

N

⁢

ln

⁡

1

ε

.

So the number of samples required to find a match with probability

ε

>

0

is approximately

k

=

2

⁢

N

⁢

1

1

-

ε

.

References

1

Wikipedia,

Birthday paradox

,

http://en.wikipedia.org/w/index.php?title=Birthday_paradox&oldid=69569553

. Accessed 14 August, 2006.

Title

approximating the birthday problem

Canonical name

ApproximatingTheBirthdayProblem

Date of creation

2013-03-22 16:09:56

Last modified on

2013-03-22 16:09:56

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

9

Author

Algeboy (12884)

Entry type

Derivation

Classification

msc 60C05

Classification

msc 05A10

Classification

msc 60-00

Generated on Sat Feb 10 11:54:36 2018 by

LaTeXML
Expectation of a non negative random variable

Expectation of a non negative random variable

For any non

negative

continuous random variable

having

distribution function

F

⁢

(

X

)

we have the followings:

1.

E

[

X

]

=

∫

0

∞

P

r

[

X

>

t

]

d

t

2.

E

[

X

r

]

=

r

∫

0

∞

t

r

-

1

P

r

[

X

>

t

]

d

t

3.

E

⁢

[

min

⁡

(

X

,

T

)

]

=

T

-

∫

0

∞

F

⁢

(

T

)

⁢

𝑑

t

4.

E

[

X

|

X

<

T

]

=

T

-

1

T

∫

0

T

F

(

t

)

d

t

where

T

is a

constant

.

Title

Expectation

of a non negative

random variable

Canonical name

ExpectationOfANonNegativeRandomVariable

Date of creation

2013-03-22 19:10:52

Last modified on

2013-03-22 19:10:52

Owner

georgiosl (7242)

Last modified by

georgiosl (7242)

Numerical id

7

Author

georgiosl (7242)

Entry type

Theorem

Classification

msc 60C05

Classification

msc 05A10

Classification

msc 60-00

Generated on Sat Feb 10 11:54:38 2018 by

LaTeXML
Buffon’s needle

Buffon’s needle

The plane is ruled by

parallel lines

2

inches apart and a

1

-inch long needle is dropped at random on the
plane. What is the

probability

that it hits parallel lines?

Solution.

The first issue is to find some appropriate

probability space

(

Ω

,

ℱ

,

P

)

. For this,

•

h

=

distance

from the center of the needle to the nearest line

•

θ

=

the angle that the needle makes with the horizontal ranging from

0

to

π

2

.

These fully determine the position of the needle. Let us next take the

1.

The probability space is

Ω

=

[

0

,

1

]

×

[

0

,

π

2

)

2.

The probability of an event

B

is denoted by

P

⁢

[

B

]

is equal to

a

⁢

r

⁢

e

⁢

a

⁢

o

⁢

f

⁢

B

π

2

Now we denote by

A

the event that the needle hits a horizontal line. It is easily seen that this happens
when

sin

⁡

θ

≥

h

1

/

2

. Consequently

A

=

{

(

θ

,

h

)

∈

Ω

:

h

≤

sin

⁡

θ

2

}

and then we get

P

⁢

[

A

]

=

2

π

⁢

∫

0

π

2

1

2

⁢

sin

⁡

θ

⁢

d

⁢

θ

=

1

π

⁢

□

In general case, when the length of needle is

l

and the distance of parallel lines is

d

provided that

l

<

d

, the probability we want is

2

⁢

l

π

⁢

d

. This is obvious just taking the

l

/

d

-point from one edge instead of the center of the needle.

Title

Buffon’s needle

Canonical name

BuffonsNeedle

Date of creation

2013-03-22 16:09:28

Last modified on

2013-03-22 16:09:28

Owner

georgiosl (7242)

Last modified by

georgiosl (7242)

Numerical id

8

Author

georgiosl (7242)

Entry type

Definition

Classification

msc 60D05

Classification

msc 60-00

Generated on Sat Feb 10 11:54:40 2018 by

LaTeXML
binomial distribution

binomial distribution

Consider an experiment with two possible outcomes (success and failure), which happen randomly. Let

p

be the

probability

of success. If the experiment is repeated

n

times, the probability of having exactly

x

successes is

f

⁢

(

x

)

=

(

n

x

)

⁢

p

x

⁢

(

1

-

p

)

(

n

-

x

)

.

The

distribution function

determined by the

probability function

f

⁢

(

x

)

is called a

Bernoulli distribution

or

binomial distribution

.

Here are some plots for

f

⁢

(

x

)

with

n

=

20

and

p

=

0.3

,

p

=

0.5

.

The corresponding distribution function is

F

⁢

(

x

)

=

∑

k

≤

x

(

n

k

)

⁢

p

k

⁢

q

n

-

k

where

q

=

1

-

p

. Notice that if we calculate

F

⁢

(

n

)

we get the binomial expansion for

(

p

+

q

)

n

, and this is the reason for the

distribution

being called

binomial

.

We will use the

moment generating function

to calculate the mean and

variance

for the distribution. The mentioned

function

is

G

⁢

(

t

)

=

∑

x

=

0

n

e

t

⁢

x

⁢

(

n

x

)

⁢

p

x

⁢

q

n

-

x

which simplifies to

G

⁢

(

t

)

=

(

p

⁢

e

t

+

q

)

n

.

Differentiating gives us

G

′

⁢

(

t

)

=

n

⁢

(

p

⁢

e

t

+

q

)

n

-

1

⁢

p

⁢

e

t

and therefore the mean is

μ

=

E

⁢

[

X

]

=

G

′

⁢

(

0

)

=

n

⁢

p

.

Now for the variance we need the second derivative

G

′′

⁢

(

t

)

=

n

⁢

(

n

-

1

)

⁢

(

p

⁢

e

t

+

q

)

n

-

2

+

n

⁢

(

p

⁢

e

t

+

q

)

n

-

1

⁢

p

⁢

e

t

so we get

E

⁢

[

X

2

]

=

G

′′

⁢

(

0

)

=

n

⁢

(

n

-

1

)

⁢

p

2

+

n

⁢

p

and finally the variance (recall

q

=

1

-

p

):

σ

2

=

E

⁢

[

X

2

]

-

E

⁢

[

X

]

2

=

n

⁢

p

⁢

q

.

For large values of

n

, the

binomial coefficients

are hard to compute, however in this cases we can use either the

Poisson distribution

or the

normal distribution

to approximate the probabilities.

Title

binomial distribution

Canonical name

BinomialDistribution

Date of creation

2013-03-22 13:03:01

Last modified on

2013-03-22 13:03:01

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

17

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 60E05

Synonym

Bernoulli distribution

Synonym

binomial

random variable

Synonym

binomial probability function

Synonym

Bernoulli random variable

Related topic

BinomialCoefficient

Related topic

BinomialTheorem

Related topic

BernoulliRandomVariable

Generated on Sat Feb 10 11:54:42 2018 by

LaTeXML
convergence in distribution

convergence in distribution

A sequence of

distribution functions

F

1

,

F

2

,

…

converges

weakly

to a distribution
function

F

if

F

n

⁢

(

t

)

→

F

⁢

(

t

)

for each point

t

at which

F

is

continuous

.

If the

random variables

X

,

X

1

,

X

2

,

…

have associated distribution functions

F

,

F

1

,

F

2

,

…

, respectively, then we say that

X

n

converges

in

distribution

to

X

, and denote this by

X

n

→

𝐷

X

.

This definition holds for

joint distribution functions

and

random vectors

as well.

This is probably the weakest of convergence of random variables. Some results involving this of convergence
are the

central limit theorems

,

Helly-Bray theorem

, Paul Lévy continuity theorem, Cramér-Wold theorem and Scheffé’s theorem.

Title

convergence in distribution

Canonical name

ConvergenceInDistribution

Date of creation

2013-03-22 13:14:12

Last modified on

2013-03-22 13:14:12

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

11

Author

Koro (127)

Entry type

Definition

Classification

msc 60E05

Related topic

WeakConvergence

Generated on Sat Feb 10 11:54:44 2018 by

LaTeXML
Cramér-Wold theorem

Cramér-Wold theorem

Let

X

¯

n

=

(

X

n

⁢

1

,

…

,

X

n

⁢

k

)

⁢

and

⁢

X

¯

=

(

X

1

,

…

,

X

k

)

be

random vectors

. Then

X

¯

n

converges to

X

¯

in

distribution

(

http://planetmath.org/ConvergenceInDistribution

) if and only if

∑

i

=

1

k

t

i

⁢

X

n

⁢

i

→

n

→

∞

𝐷

∑

i

=

1

k

t

i

⁢

X

i

.

for each

(

t

1

,

…

,

t

k

)

∈

ℝ

k

. That is, if every

linear combination

of the

coordinates

of

X

¯

n

converges in distribution to the correspondent linear combination of coordinates of

X

¯

.

Title

Cramér-Wold theorem

Canonical name

CramerWoldTheorem

Date of creation

2013-03-22 13:14:21

Last modified on

2013-03-22 13:14:21

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

4

Author

Koro (127)

Entry type

Theorem

Classification

msc 60E05

Generated on Sat Feb 10 11:54:46 2018 by

LaTeXML
cumulant generating function

cumulant generating function

Given a

random variable

X

, the

cumulant generating function

of

X

is the following function:

H

X

⁢

(

t

)

=

ln

⁡

E

⁢

[

e

t

⁢

X

]

for all

t

∈

R

in which the

expectation

converges.

In other , the cumulant generating function is just the logarithm of the

moment generating function

.

The cumulant generating function of

X

is defined on a (possibly degenerate) interval containing

t

=

0

; one has

H

X

⁢

(

0

)

=

0

; moreover,

H

X

⁢

(

t

)

is a convex function (

http://planetmath.org/ConvexFunction

). (Indeed, the moment generating function is defined on a possibly degenerate interval containing

t

=

0

, which image is a

positive

interval containing

t

=

1

; so the logarithm is defined on the same interval on which is defined the moment generating function.)

The

k

th-derivative of the cumulant generating function evaluated at zero is the

k

th

cumulant

of

X

.

Title

cumulant generating function

Canonical name

CumulantGeneratingFunction

Date of creation

2013-03-22 16:16:24

Last modified on

2013-03-22 16:16:24

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

17

Author

Andrea Ambrosio (7332)

Entry type

Definition

Classification

msc 60E05

Related topic

MomentGeneratingFunction

Related topic

CharacteristicFunction2

Generated on Sat Feb 10 11:54:47 2018 by

LaTeXML
degrees of freedom

degrees of freedom

The number of

degrees of freedom

of a

distribution

(

http://planetmath.org/Distribution3

) is the number of

parameters

that are used to the value of the distribution and are either known or assumed to be

independent

.

The following use degrees of freedom:

•

chi-squared random variable

•

chi-squared statistic

•

F distribution

•

non-central chi-squared random variable

•

t distribution

Remark:

The number of

degrees of freedom

has also a physical meaning
related to the possible directions of motion of the system.

Title

degrees of freedom

Canonical

name

DegreesOfFreedom

Date of creation

2013-03-22 17:10:53

Last modified on

2013-03-22 17:10:53

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

7

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 60E05

Classification

msc 62A01

Generated on Sat Feb 10 11:54:50 2018 by

LaTeXML