aarjaneiro/planetmath huggingface dataset:
density function

density function

Let

X

be a

discrete random variable

with

sample space

{

x

1

,

x

2

,

…

}

.
Let

p

k

be the

probability

of

X

taking the value

x

k

.

The

function

f

⁢

(

x

)

=

{

p

k

if

⁢

x

=

x

k

0

otherwise

is called the

probability function

or

density function

.

It must hold:

∑

j

=

1

∞

f

⁢

(

x

j

)

=

1

If the density function for a

random variable

is known, we can calculate the probability of

X

being on certain interval:

P

[

a

<

X

≤

b

]

=

∑

a

<

x

j

≤

b

f

(

x

j

)

=

∑

a

<

x

j

≤

b

p

j

.

The definition can be extended to continuous random variables in a direct way: The probability of

x

being on a given interval is calculated with an

integral

instead of using a

summation

:

P

[

a

<

X

≤

b

]

=

∫

a

b

f

(

x

)

d

x

.

For a more formal approach using measure theory, look at

probability distribution function

entry.

Title

density function

Canonical name

DensityFunction

Date of creation

2013-03-22 13:02:49

Last modified on

2013-03-22 13:02:49

Owner

drini (3)

Last modified by

drini (3)

Numerical id

12

Author

drini (3)

Entry type

Definition

Classification

msc 60E05

Synonym

probability function

Synonym

density

Synonym

probabilities function

Related topic

DistributionFunction

Related topic

CumulativeDistributionFunction

Related topic

RandomVariable

Related topic

Distribution

Related topic

GeometricDistribution2

Generated on Sat Feb 10 11:54:52 2018 by

LaTeXML
distribution function

distribution function

[this entry is currently being revised, so hold off on corrections until
this line is removed]

Let

F

:

ℝ

→

ℝ

. Then

F

is a

distribution function

if

1.

F

is nondecreasing,

2.

F

is

continuous

from the right,

3.

lim

x

→

-

∞

⁡

F

⁢

(

x

)

=

0

, and

lim

x

→

∞

⁡

F

⁢

(

x

)

=

1

.

As an example, suppose that

Ω

=

ℝ

and that

ℬ

is the

σ

-

algebra

of Borel subsets of

ℝ

.
Let

P

be a

probability measure

on

(

Ω

,

ℬ

)

.
Define

F

by

F

⁢

(

x

)

=

P

⁢

(

(

-

∞

,

x

]

)

.

This particular

F

is called the

distribution function

of

P

. It is
easy to verify that 1,2, and 3 hold for this

F

.

In fact, every distribution function is the distribution function of some
probability measure on the Borel subsets of

ℝ

. To see this,
suppose that

F

is a distribution function. We can define

P

on a single half-open
interval by

P

⁢

(

(

a

,

b

]

)

=

F

⁢

(

b

)

-

F

⁢

(

a

)

and extend

P

to unions of disjoint intervals by

P

⁢

(

∪

i

=

1

∞

(

a

i

,

b

i

]

)

=

∑

i

=

1

∞

P

⁢

(

(

a

i

,

b

i

]

)

.

and then further extend

P

to all the Borel subsets of

ℝ

.
It is clear that the distribution function of

P

is

F

.

0.1

Random Variables

Suppose that

(

Ω

,

ℬ

,

P

)

is a

probability space

and

X

:

Ω

→

ℝ

is a

random variable

. Then there is an

induced

probability measure

P

X

on

ℝ

defined as
follows:

P

X

⁢

(

E

)

=

P

⁢

(

X

-

1

⁢

(

E

)

)

for every Borel subset

E

of

ℝ

.

P

X

is called the

distribution

of

X

. The

distribution function

of

X

is

F

X

(

x

)

=

P

(

ω

|

X

(

ω

)

≤

x

)

.

The distribution function of

X

is also known as the law of

X

.
Claim:

F

X

= the distribution function of

P

X

.

F

X

⁢

(

x

)

=

P

(

ω

|

X

(

ω

)

≤

x

)

=

P

(

X

-

1

(

(

-

∞

,

x

]

)

=

P

X

⁢

(

(

-

∞

,

x

]

)

=

F

⁢

(

x

)

.

0.2

Density Functions

Suppose that

f

:

ℝ

→

ℝ

is a nonnegative function
such that

∫

-

∞

∞

f

⁢

(

t

)

⁢

𝑑

t

=

1

.

Then one can define

F

:

ℝ

→

ℝ

by

F

⁢

(

x

)

=

∫

-

∞

x

f

⁢

(

t

)

⁢

𝑑

t

.

Then it is clear that

F

satisfies the conditions 1,2,and 3 so

F

is a distribution function. The function

f

is called a

density function

for the distribution

F

.

If

X

is a discrete random variable with density function

f

and distribution
function

F

then

F

⁢

(

x

)

=

∑

x

j

≤

x

f

⁢

(

x

j

)

.

Title

distribution function

Canonical name

DistributionFunction

Date of creation

2013-03-22 13:02:51

Last modified on

2013-03-22 13:02:51

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

16

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 60E05

Synonym

cumulative distribution function

Synonym

distribution

Related topic

DensityFunction

Related topic

CumulativeDistributionFunction

Related topic

RandomVariable

Related topic

Distribution

Related topic

GeometricDistribution2

Defines

law of a random variable

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

LaTeXML
geometric distribution

geometric distribution

Suppose that a random experiment has two possible outcomes, success with

probability

p

and failure with probability

q

=

1

-

p

. The experiment is repeated until a success happens. The number of trials before the success is a

random variable

X

with

density function

f

⁢

(

x

)

=

q

(

x

-

1

)

⁢

p

.

The

distribution function

determined by

f

⁢

(

x

)

is called a

geometric distribution

with parameter

p

and it is given by

F

⁢

(

x

)

=

∑

k

≤

x

q

(

k

-

1

)

⁢

p

.

The picture shows the graph for

f

⁢

(

x

)

with

p

=

1

/

4

. Notice the quick

decreasing

. An

interpretation

is that a long run of failures is very unlikely.

We can use the

moment generating function

method in order to get the mean and

variance

. This

function

is

G

⁢

(

t

)

=

∑

k

=

1

∞

e

t

⁢

k

⁢

q

(

k

-

1

)

⁢

p

=

p

⁢

e

t

⁢

∑

k

=

0

∞

(

e

t

⁢

q

)

k

.

The last expression can be simplified as

G

⁢

(

t

)

=

p

⁢

e

t

1

-

e

t

⁢

q

.

The

first derivative

is

G

′

⁢

(

t

)

=

e

t

⁢

p

(

1

-

e

t

⁢

q

)

2

so the mean is

μ

=

E

⁢

[

X

]

=

G

′

⁢

(

0

)

=

1

p

.

In order to find the variance, we use the second derivative and thus

E

⁢

[

X

2

]

=

G

′′

⁢

(

0

)

=

2

-

p

p

2

and therefore the variance is

σ

2

=

E

⁢

[

X

2

]

-

E

⁢

[

X

]

2

=

G

′′

⁢

(

0

)

-

G

′

⁢

(

0

)

2

=

q

p

2

.

Title

geometric distribution

Canonical name

GeometricDistribution

Date of creation

2013-03-22 13:03:07

Last modified on

2013-03-22 13:03:07

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

14

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 60E05

Synonym

geometric random variable

Related topic

RandomVariable

Related topic

DensityFunction

Related topic

DistributionFunction

Related topic

Mean

Related topic

Variance

Related topic

BernoulliDistribution

Related topic

ArithmeticMean

Generated on Sat Feb 10 11:54:57 2018 by

LaTeXML
Gumbel random variable

Gumbel random variable

X

is a

Gumbel random variable

if it has a

probability density function

, given by

f

X

⁢

(

x

)

=

1

σ

⁢

exp

⁡

(

x

-

μ

σ

)

⁢

S

⁢

(

x

)

where

-

∞

<

x

<

∞

,

μ

is the

location parameter

,

σ

is the

scale parameter

, and

S

⁢

(

x

)

is the

survivor function

,

S

⁢

(

x

)

=

exp

⁡

[

-

exp

⁡

(

x

-

μ

σ

)

]

.

Notation for

X

having a

Gumbel distribution

is

X

∼

Gum

⁢

(

μ

,

σ

)

.

:
Given a Gumbel distribution

X

∼

Gum

⁢

(

μ

,

σ

)

:

1.

E[X]=

μ

-

γ

⁢

σ

, where

γ

is the Euler’s constant

2.

Var[X]=

π

2

6

⁢

σ

2

Remark.

Nevertheless the interval

(

-

∞

,

∞

)

in which is defined, the Gumbel distribution is often used to model reliability or lifetime of products.

Title

Gumbel random variable

Canonical name

GumbelRandomVariable

Date of creation

2013-03-22 15:55:40

Last modified on

2013-03-22 15:55:40

Owner

georgiosl (7242)

Last modified by

georgiosl (7242)

Numerical id

4

Author

georgiosl (7242)

Entry type

Definition

Classification

msc 60E05

Generated on Sat Feb 10 11:54:58 2018 by

LaTeXML
Helly-Bray theorem

Helly-Bray theorem

Let

F

,

F

1

,

F

2

,

…

be

distribution functions

. If

F

n

converges

weakly (

http://planetmath.org/ConvergenceInDistribution

) to

F

, then

∫

ℝ

g

⁢

(

x

)

⁢

𝑑

F

n

⁢

(

x

)

→

n

→

∞

∫

ℝ

g

⁢

(

x

)

⁢

𝑑

F

⁢

(

x

)

for each

continuous

bounded function

g

:

ℝ

→

ℝ

.

Remark.

The

integrals

involved are

Riemann-Stieltjes integrals

.

Title

Helly-Bray theorem

Canonical name

HellyBrayTheorem

Date of creation

2013-03-22 13:14:18

Last modified on

2013-03-22 13:14:18

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

6

Author

Koro (127)

Entry type

Theorem

Classification

msc 60E05

Generated on Sat Feb 10 11:55:00 2018 by

LaTeXML
joint discrete density function

joint discrete density function

Let

X

1

,

X

2

,

…

,

X

n

be

n

random variables

all defined on the same

probability space

. The

joint discrete density function

of

X

1

,

X

2

,

…

,

X

n

, denoted by

f

X

1

,

X

2

,

…

,

X

n

⁢

(

x

1

,

x

2

,

…

,

x

n

)

, is the following

function

:

f

X

1

,

X

2

,

…

,

X

n

:

R

n

→

R

f

X

1

,

X

2

,

…

,

X

n

(

x

1

,

x

2

,

…

,

x

n

)

=

P

[

X

1

=

x

1

,

X

2

=

x

2

,

…

,

X

n

=

x

n

]

As in the single

variable

case, sometimes it’s expressed as

p

X

1

,

X

2

,

…

,

X

n

⁢

(

x

1

,

x

2

,

…

,

x

n

)

to mark the

difference

between this function and the

continuous

joint density function

.

Also, as in the case where

n

=

1

, this function

satisfies

:

1.

f

X

1

,

X

2

,

…

,

X

n

⁢

(

x

1

,

…

,

x

n

)

≥

0

∀

(

x

1

,

…

,

x

n

)

2.

∑

x

1

,

…

,

x

n

f

X

1

,

X

2

,

…

,

X

n

⁢

(

x

1

,

…

,

x

n

)

=

1

In this case,

f

X

1

,

X

2

,

…

,

X

n

(

x

1

,

…

,

x

n

)

=

P

[

X

1

=

x

1

,

X

2

=

x

2

,

…

,

X

n

=

x

n

]

.

Title

joint discrete density function

Canonical

name

JointDiscreteDensityFunction

Date of creation

2013-03-22 11:54:55

Last modified on

2013-03-22 11:54:55

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

10

Author

mathcam (2727)

Entry type

Definition

Classification

msc 60E05

Synonym

joint probability function

Synonym

joint distribution

Generated on Sat Feb 10 11:55:02 2018 by

LaTeXML
marginal distribution

marginal distribution

Given

random variables

X

1

,

X

2

,

…

,

X

n

and a subset

I

⊂

{

1

,

2

,

…

,

n

}

, the

marginal distribution

of the random variables

X

i

:

i

∈

I

is the following:

f

{

X

i

:

i

∈

I

}

⁢

(

𝐱

)

=

∑

{

x

i

:

i

∉

I

}

f

X

1

,

…

,

X

n

⁢

(

x

1

,

…

,

x

n

)

or

f

{

X

i

:

i

∈

I

}

⁢

(

𝐱

)

=

∫

{

x

i

:

i

∉

I

}

f

X

1

,

…

,

X

n

⁢

(

u

1

,

…

,

u

n

)

⁢

∏

{

u

i

:

i

∉

I

}

d

⁢

u

i

,

summing

if the variables are discrete and integrating if the variables are

continuous

.

This is, the marginal distribution of a set of random variables

X

1

,

…

,

X

n

can be obtained by summing (or integrating) the

joint distribution

over all values of the other variables.

The most common marginal distribution is the

individual

marginal distribution (ie, the marginal distribution of ONE random variable).

Title

marginal distribution

Canonical name

MarginalDistribution

Date of creation

2013-03-22 11:55:00

Last modified on

2013-03-22 11:55:00

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

10

Author

mathcam (2727)

Entry type

Algorithm

Classification

msc 60E05

Synonym

marginal density function

Synonym

marginal probability function

Generated on Sat Feb 10 11:55:04 2018 by

LaTeXML
moment generating function

moment generating function

Given a

random variable

X

, the

moment generating function

of

X

is the following function:

M

X

⁢

(

t

)

=

E

⁢

[

e

t

⁢

X

]

for

t

∈

R

(if the

expectation

converges).

It can be shown that if the moment generating function of

X

is defined on an interval around the origin, then

E

⁢

[

X

k

]

=

M

X

(

k

)

⁢

(

t

)

|

t

=

0

In other words, the

k

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

k

th moment of

X

.

Title

moment generating function

Canonical name

MomentGeneratingFunction

Date of creation

2013-03-22 11:53:51

Last modified on

2013-03-22 11:53:51

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

10

Author

mathcam (2727)

Entry type

Definition

Classification

msc 60E05

Classification

msc 46L05

Classification

msc 82-00

Classification

msc 83-00

Classification

msc 81-00

Related topic

CharacteristicFunction2

Related topic

CumulantGeneratingFunction

Generated on Sat Feb 10 11:55:06 2018 by

LaTeXML
moment generating function of the sum of independent random variables

moment generating function of the sum of independent random variables

Let

X

i

be

independent

random variables

for

i

=

1

,

…

,

n

, let each

X

i

have

moment generating function

M

X

i

⁢

(

t

)

, and let

X

=

∑

i

=

1

n

X

i

. Then the moment generating function of

X

is

M

X

⁢

(

t

)

=

∏

i

=

1

n

M

X

i

⁢

(

t

)

.

Proof.

By

definition

,

M

X

⁢

(

t

)

=

E

⁢

(

e

t

⁢

X

)

=

E

⁢

(

e

t

⁢

(

X

1

+

⋯

+

X

n

)

)

=

E

⁢

(

e

t

⁢

X

1

⁢

⋯

⁢

e

t

⁢

X

n

)

.

Now, since each

X

i

is independent of the others, this becomes

M

X

⁢

(

t

)

=

E

⁢

(

e

t

⁢

X

1

)

⁢

⋯

⁢

E

⁢

(

e

t

⁢

X

n

)

=

M

X

1

⁢

(

t

)

⁢

⋯

⁢

M

X

n

⁢

(

t

)

=

∏

i

=

1

n

M

X

i

⁢

(

t

)

as required.
∎

Title

moment generating function of the sum of independent random variables

Canonical

name

MomentGeneratingFunctionOfTheSumOfIndependentRandomVariables

Date of creation

2013-03-22 17:17:11

Last modified on

2013-03-22 17:17:11

Owner

me_and (17092)

Last modified by

me_and (17092)

Numerical id

5

Author

me_and (17092)

Entry type

Corollary

Classification

msc 60E05

Generated on Sat Feb 10 11:55:09 2018 by

LaTeXML
multinomial distribution

multinomial distribution

Let

𝐗

=

(

X

1

,

…

,

X

n

)

be a

random vector

such that

1.

X

i

≥

0

and

X

i

∈

ℤ

2.

X

1

+

⋯

+

X

n

=

N

, where

N

is a fixed integer

Then

X

has a

multinomial distribution

if there exists a

parameter

vector

𝝅

=

(

π

1

,

…

,

π

n

)

such that

1.

π

i

≥

0

and

π

i

∈

ℝ

2.

π

1

+

⋯

+

π

n

=

1

3.

X

has a discrete

probability distribution function

f

𝐗

⁢

(

𝒙

)

in the form:

f

𝐗

⁢

(

𝒙

)

=

N

!

x

1

!

⁢

⋯

⁢

x

n

!

⁢

∏

i

=

1

n

π

i

x

i

Remarks

•

E

⁡

[

𝐗

]

=

N

⁢

𝝅

•

Var

⁡

[

𝐗

]

=

(

v

i

⁢

j

)

, where

v

i

⁢

j

=

{

N

⁢

π

i

⁢

(

1

-

π

i

)

if

i

=

j

;

-

N

⁢

π

i

⁢

π

j

if

i

≠

j

.

•

When

n

=

2

, the multinomial distribution is the same as the

binomial distribution

•

If

X

1

,

…

,

X

n

are mutually

independent

Poisson random variables

with parameters

λ

1

,

…

,

λ

n

respectively, then the

conditional

joint distribution

of

X

1

,

…

,

X

n

given that

X

1

+

⋯

+

X

n

=

N

is
multinomial with parameters

λ

i

/

λ

, where

λ

=

∑

λ

i

.

Sketch of proof.

Each

X

i

is distributed as:

f

X

i

⁢

(

x

i

)

=

e

-

λ

i

⁢

λ

i

x

i

x

i

!

The mutual independence of the

X

i

’s shows that the joint probability distribution of the

X

i

’s is given by

f

𝐗

⁢

(

𝒙

)

=

∏

i

=

1

n

e

-

λ

i

⁢

λ

i

x

i

x

i

!

=

e

-

λ

⁢

∏

i

=

1

n

λ

i

x

i

x

i

!

,

where

𝐗

=

(

X

1

,

…

,

X

n

)

,

𝒙

=

(

x

1

,

…

,

x

n

)

and

λ

=

λ

1

+

⋯

+

λ

n

.
Next, let

X

=

X

1

+

⋯

+

X

n

. Then

X

is Poisson distributed with parameter

λ

(which can be shown by using

induction

and the mutual independence of the

X

i

’s):

f

X

⁢

(

x

)

=

e

-

λ

⁢

λ

x

x

!

.

The conditional probability distribution of

X

given that

X

=

N

is thus given by:

f

𝐗

(

𝒙

∣

X

=

N

)

=

f

𝐗

⁢

(

𝒙

)

f

X

⁢

(

N

)

=

(

e

-

λ

∏

i

=

1

n

λ

i

x

i

x

i

!

)

/

(

e

-

λ

⁢

λ

N

N

!

)

=

N

!

x

1

!

⁢

⋯

⁢

x

n

!

∏

i

=

1

n

(

λ

i

λ

)

x

i

,

where

∑

x

i

=

N

and that

∑

λ

i

/

λ

=

1

.

Title

multinomial distribution

Canonical

name

MultinomialDistribution

Date of creation

2013-03-22 14:33:35

Last modified on

2013-03-22 14:33:35

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 60E05

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

LaTeXML
multivariate distribution function

multivariate distribution function

A function

F

:

ℝ

n

→

[

0

,

1

]

is said to be a

multivariate distribution function

if

1.

F

is non-decreasing in each of its arguments; i.e., for any

1

≤

i

≤

n

, the function

G

i

:

ℝ

→

[

0

,

1

]

given by

G

i

⁢

(

x

)

:=

F

⁢

(

a

1

,

…

,

a

i

-

1

,

x

,

a

i

+

1

,

…

,

a

n

)

is non-decreasing for any set of

a

j

∈

ℝ

such that

j

≠

i

.

2.

G

i

⁢

(

-

∞

)

=

0

, where

G

i

is defined as above; i.e., the limit of

G

i

as

x

→

-

∞

is

0

3.

F

⁢

(

∞

,

…

,

∞

)

=

1

; i.e. the limit of

F

as each of its arguments approaches infinity, is 1.

Generally, right-continuty of

F

in each of its arguments is added as one of the conditions, but it is not assumed here.

If, in the second condition above, we set

a

j

=

∞

for

j

≠

i

, then

G

i

⁢

(

x

)

is called a (one-dimensional)

margin

of

F

. Similarly, one defines an

m

-dimensional (

m

<

n

)

margin

of

F

by setting

n

-

m

of the arguments in

F

to

∞

. For each

m

<

n

, there are

(

n

m

)

m

-dimensional margins of

F

. Each

m

-dimensional margin of a multivariate distribution function is itself a multivariate distribution function. A one-dimensional margin is a

distribution function

.

Multivariate distribution functions are typically found in probability theory, and especially in

statistics

. An example of a commonly used multivariate distribution function is the

multivariate Gaussian distribution

function. In

ℝ

2

, the standard bivariate

Gaussian distribution

function (with zero

mean vector

, and the

identity matrix

as its

covariance matrix

) is given by

F

⁢

(

x

,

y

)

=

1

2

⁢

π

⁢

∫

-

∞

x

∫

-

∞

y

exp

⁡

(

-

s

2

+

t

2

2

)

⁢

𝑑

s

⁢

𝑑

t

B. Schweizer and A. Sklar have generalized the above definition to include a wider class of functions. The generalization has to do with the weakening of the coordinate-wise non-decreasing condition (first condition above). The attempt here is to study a class of functions that can be used as models for

distributions

of distances between points in a “probabilistic metric space”.

References

1

B. Schweizer and A. Sklar,

Probabilistic Metric Spaces

, Dover Publications, (2005).

Title

multivariate distribution function

Canonical name

MultivariateDistributionFunction

Date of creation

2013-03-22 16:33:50

Last modified on

2013-03-22 16:33:50

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 60E05

Classification

msc 62E10

Related topic

Copula

Defines

multivariate cumulative distribution function

Defines

joint distribution function

Defines

margin

Generated on Sat Feb 10 11:55:21 2018 by

LaTeXML
relative entropy

relative entropy

Let

p

and

q

be probability distributions with

supports

𝒳

and

𝒴

respectively, where

𝒳

⊂

𝒴

. The

relative entropy

or

Kullback-Leibler

distance

between two probability distributions

p

and

q

is defined as

D

(

p

|

|

q

)

:=

∑

x

∈

𝒳

p

(

x

)

log

p

⁢

(

x

)

q

⁢

(

x

)

.

(1)

While

D

(

p

|

|

q

)

is often called a distance, it is not a true metric because it is not

symmetric

and does not

satisfy

the

triangle inequality

. However, we do have

D

(

p

|

|

q

)

≥

0

with equality iff

p

=

q

.

-

D

(

p

|

|

q

)

=

-

∑

x

∈

𝒳

p

⁢

(

x

)

⁢

log

⁡

p

⁢

(

x

)

q

⁢

(

x

)

(2)

=

∑

x

∈

𝒳

p

⁢

(

x

)

⁢

log

⁡

q

⁢

(

x

)

p

⁢

(

x

)

(3)

≤

log

⁡

(

∑

x

∈

𝒳

p

⁢

(

x

)

⁢

q

⁢

(

x

)

p

⁢

(

x

)

)

(4)

=

log

⁡

(

∑

x

∈

𝒳

q

⁢

(

x

)

)

(5)

≤

log

⁡

(

∑

x

∈

𝒴

q

⁢

(

x

)

)

(6)

=

0

(7)

where the first

inequality

follows from the concavity of

log

⁡

(

x

)

and the second from expanding the sum over the support of

q

rather than

p

.

Relative entropy also comes in a

continuous

version which looks just as one might expect. For

continuous distributions

f

and

g

,

𝒮

the support of

f

, we have

D

(

f

|

|

g

)

:=

∫

𝒮

f

log

f

g

.

(8)

Title

relative entropy

Canonical name

RelativeEntropy

Date of creation

2013-03-22 12:19:32

Last modified on

2013-03-22 12:19:32

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

10

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 60E05

Classification

msc 94A17

Synonym

Kullback-Leibler distance

Related topic

Metric

Related topic

ConditionalEntropy

Related topic

MutualInformation

Related topic

ProofOfGaussianMaximizesEntropyForGivenCovariance

Generated on Sat Feb 10 11:55:23 2018 by

LaTeXML
Scheffé’s theorem

Scheffé’s theorem

Let

X

,

X

1

,

X

2

,

…

be

continuous random variables

in a

probability space

, whose

probability density functions

are

f

,

f

1

,

f

2

,

…

, respectively. If

f

n

→

f

almost everywhere (relative to

Lebesgue measure

,) then

X

n

converges

to

X

in

distribution

(

http://planetmath.org/ConvergenceInDistribution

):

X

n

→

𝐷

X

.

Title

Scheffé’s

theorem

Canonical name

ScheffesTheorem

Date of creation

2013-03-22 13:14:23

Last modified on

2013-03-22 13:14:23

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

6

Author

Koro (127)

Entry type

Theorem

Classification

msc 60E05

Generated on Sat Feb 10 11:55:25 2018 by

LaTeXML
standard deviation

standard deviation

Given a

random variable

X

, the

standard deviation

of

X

is defined as

S

⁢

D

⁢

[

X

]

=

Var

⁡

[

X

]

.

The standard deviation is a

measure

of the variation of

X

around the

expected value

.

Title

standard deviation

Canonical name

StandardDeviation

Date of creation

2013-03-22 11:53:49

Last modified on

2013-03-22 11:53:49

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Definition

Classification

msc 60E05

Classification

msc 46L05

Classification

msc 82-00

Classification

msc 83-00

Classification

msc 81-00

Classification

msc 22A22

Related topic

Variance

Generated on Sat Feb 10 11:55:26 2018 by

LaTeXML