aarjaneiro/planetmath huggingface dataset:
expected value

expected value

Let us first consider a

discrete random variable

X

with values in

ℝ

. Then

X

has values in an at most

countable set

𝒳

. For

x

∈

𝒳

denote the

probability

that

X

=

x

by

P

x

. If

∑

x

∈

𝒳

|

x

|

⁢

P

x

converges

, the sum

∑

x

∈

𝒳

x

⁢

P

x

is well-defined. Its value is called the

expected value

,

expectation

or

mean

of

X

. It is usually denoted by

E

⁢

(

X

)

.

Taking this idea further, we can easily generalize to a continuous random variable

X

with probability density

ϱ

by setting

E

⁢

(

X

)

=

∫

-

∞

∞

x

⁢

ϱ

⁢

(

x

)

⁢

𝑑

x

,

if this integral exists.

From the above definition it is clear that the expectation is a

linear function

, i.e. for two

random variables

X

,

Y

we have

E

⁢

(

a

⁢

X

+

b

⁢

Y

)

=

a

⁢

E

⁢

(

X

)

+

b

⁢

E

⁢

(

Y

)

for

a

,

b

∈

ℝ

.

Note that the expectation does not always exist (if the corresponding sum or integral does not converge, the expectation does not exist. One example of this situation is the

Cauchy random variable

).

Using the

measure

theoretical formulation of stochastics, we can give a more formal definition. Let

(

Ω

,

𝒜

,

P

)

be a

probability space

and

X

:

Ω

→

ℝ

a random variable. We now define

E

⁢

(

X

)

=

∫

Ω

X

⁢

𝑑

P

,

where the integral is understood as the Lebesgue-integral with respect to the measure

P

.

Title

expected value

Canonical name

ExpectedValue

Date of creation

2013-03-22 11:53:42

Last modified on

2013-03-22 11:53:42

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

21

Author

mathwizard (128)

Entry type

Definition

Classification

msc 60-00

Classification

msc 46L05

Classification

msc 82-00

Classification

msc 83-00

Classification

msc 81-00

Synonym

mean

Synonym

expectation value

Synonym

expectation

Related topic

AverageValueOfFunction

Generated on Sat Feb 10 11:51:57 2018 by

LaTeXML
gamma random variable

gamma random variable

A

gamma random variable

with

parameters

α

>

0

and

λ

>

0

is one whose

probability density function

is given by

f

X

⁢

(

x

)

=

λ

α

Γ

⁢

(

α

)

⁢

x

α

-

1

⁢

e

-

λ

⁢

x

for

x

>

0

, and is denoted by

X

∼

G

⁢

a

⁢

m

⁢

m

⁢

a

⁢

(

α

,

λ

)

.

Notes:

1.

Gamma random variables are widely used in many applications. Taking

α

=

1

reduces the form to that of an

exponential random variable

. If

α

=

n

2

and

λ

=

1

2

, this is a

chi-squared random variable

.

2.

The function

Γ

:

[

0

,

∞

]

→

R

is the gamma function, defined as

Γ

⁢

(

t

)

=

∫

0

∞

x

t

-

1

⁢

e

-

x

⁢

𝑑

x

.

3.

The

expected value

of a gamma random variable is given by

E

⁢

[

X

]

=

α

λ

, and the

variance

by

V

⁢

a

⁢

r

⁢

[

X

]

=

α

λ

2

4.

The

moment generating function

of a gamma random variable is given by

M

X

⁢

(

t

)

=

(

λ

λ

-

t

)

α

.

If the first parameter is a

positive integer

, the

variate

is usually called Erlang

random variate

. The sum of

n

exponentially distributed

variables

with parameter

λ

is a gamma (Erlang) variate with parameters

n

,

λ

.

Title

gamma random variable

Canonical

name

GammaRandomVariable

Date of creation

2013-03-22 11:54:27

Last modified on

2013-03-22 11:54:27

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

14

Author

mathcam (2727)

Entry type

Definition

Classification

msc 60-00

Classification

msc 62-00

Synonym

gamma distribution

Defines

Erlang random variable

Generated on Sat Feb 10 11:51:59 2018 by

LaTeXML
independent identically distributed

independent identically distributed

Two

random variables

X

and

Y

are said to be

identically distributed

if they are defined on the same

probability space

(

Ω

,

ℱ

,

P

)

, and the

distribution function

F

X

of

X

and the distribution function

F

Y

of

Y

are the same:

F

X

=

F

Y

. When

X

and

Y

are identically distributed, we write

X

=

d

Y

.

A set of random variables

X

i

,

i

in some

index set

I

, is identically distributed if

X

i

=

d

X

j

for every pair

i

,

j

∈

I

.

A

collection

of random variables

X

i

(

i

∈

I

) is said to be

independent identically distributed

, if the

X

i

’s are identically distributed, and mutually

independent

(

http://planetmath.org/Independent

) (every finite subfamily of

X

i

is independent). This is often abbreviated as

iid

.

For example, the interarrival times

T

i

of a

Poisson process

of rate

λ

are independent and each have an

exponential distribution

with mean

1

/

λ

, so the

T

i

are independent identically distributed random variables.

Many other examples are found in

statistics

, where

individual

data points are often assumed to realizations of iid random variables.

Title

independent identically distributed

Canonical

name

IndependentIdenticallyDistributed

Date of creation

2013-03-22 14:27:29

Last modified on

2013-03-22 14:27:29

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 60-00

Synonym

iid

Synonym

independent and identically distributed

Defines

identically distributed

Generated on Sat Feb 10 11:52:01 2018 by

LaTeXML
list of probability distributions

list of probability distributions

discrete distributions

1.

Bernoulli (

http://planetmath.org/BernoulliRandomVariable

)

2.

Uniform (

http://planetmath.org/UniformDiscreteRandomVariable

)

3.

Binomial (

http://planetmath.org/BernoulliDistribution2

)

4.

Multinomial (

http://planetmath.org/MultinomialDistribution

)

5.

Negative binomial (

http://planetmath.org/NegativeBinomialRandomVariable

)

6.

Poisson (

http://planetmath.org/PoissonRandomVariable

)

7.

Geometric (

http://planetmath.org/GeometricDistribution2

) (also here (

http://planetmath.org/GeometricRandomVariable

))

8.

Hypergeometric (

http://planetmath.org/HypergeometricRandomVariable

)

9.

Negative hypergeometric (

http://planetmath.org/NegativeHypergeometricRandomVariable

)

10.

Dirac measure

(

http://planetmath.org/DiracMeasure

)

continuous distributions

1.

Uniform (

http://planetmath.org/UniformContinousRandomVariable

)

2.

Normal (

http://planetmath.org/NormalRandomVariable

)

3.

Joint normal (

http://planetmath.org/JointNormalDistribution

)

4.

Gamma (

http://planetmath.org/GammaRandomVariable

)

5.

Chi-squared (

http://planetmath.org/ChiSquaredRandomVariable

)

6.

Non-central chi-squared (

http://planetmath.org/NonCentralChiSquaredRandomVariable

)

7.

Exponential

(

http://planetmath.org/ExponentialRandomVariable

)

8.

Cauchy (

http://planetmath.org/CauchyRandomVariable

)

9.

Lognormal (

http://planetmath.org/LognormalRandomVariable

)

10.

F

distribution

(

http://planetmath.org/FDistribution

)

11.

t distribution (

http://planetmath.org/TDistribution

)

12.

Beta (

http://planetmath.org/BetaRandomVariable

)

13.

Pareto (

http://planetmath.org/ParetoRandomVariable

)

14.

Wishart (

http://planetmath.org/WishartDistribution

)

15.

Weibull (

http://planetmath.org/WeibullRandomVariable

)

16.

Gumbel (

http://planetmath.org/GumbelRandomVariable

)

Title

list of probability distributions

Canonical name

ListOfProbabilityDistributions

Date of creation

2013-03-22 18:44:26

Last modified on

2013-03-22 18:44:26

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

6

Author

gel (22282)

Entry type

Topic

Classification

msc 60-00

Classification

msc 62-00

Generated on Sat Feb 10 11:52:03 2018 by

LaTeXML
moment

moment

Moments

Given a

random variable

X

, the

k

th moment

of

X

is the value

E

⁢

[

X

k

]

, if the

expectation

exists.

Note that the

expected value

is the first moment of a random variable, and the

variance

is the second moment minus the first moment squared.

The

k

th moment of

X

is usually obtained by using the

moment generating function

.

Central moments

Given a random variable

X

, the

k

th central moment

of

X

is the value

E

⁢

[

(

X

-

E

⁢

[

X

]

)

k

]

, if the expectation exists. It is denoted by

μ

k

.

Note that the

μ

1

=

0

and

μ

2

=

V

⁢

a

⁢

r

⁢

[

X

]

=

σ

2

. The third central moment divided by the

standard deviation

cubed is called the

skewness

τ

:

τ

=

μ

3

σ

3

The skewness

measures

how “symmetrical”, or rather, how “skewed”, a

distribution

is with respect to its mode. A non-zero

τ

means there is some degree of skewness in the distribution. For example,

τ

>

0

means that the distribution has a longer

positive

tail.

The fourth central moment divided by the fourth power of the standard deviation is called the

kurtosis

κ

:

κ

=

μ

4

σ

4

The kurtosis measures how “peaked” a distribution is compared to the

standard normal distribution

. The standard normal distribution has

κ

=

3

.

κ

<

3

means that the distribution is “flatter” than then standard normal distribution, or

platykurtic

. On the other hand, a distribution with

κ

>

3

can be characterized as being more “peaked” than

N

⁢

(

0

,

1

)

, or

leptokurtic

.

Title

moment

Canonical

name

Moment

Date of creation

2013-03-22 11:53:54

Last modified on

2013-03-22 11:53:54

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 60-00

Classification

msc 62-00

Classification

msc 81-00

Defines

central moment

Defines

skewness

Defines

kurtosis

Defines

platykurtic

Defines

leptokurtic

Generated on Sat Feb 10 11:52:05 2018 by

LaTeXML
proof of Bayes’ Theorem

proof of Bayes’ Theorem

The proof of Bayes’ theorem is no more than an exercise in substituting the definition of

conditional probability

into the formula, and applying the

total probability theorem

.

ℙ

⁢

{

B

}

⁢

ℙ

⁢

{

E

|

B

}

∑

i

ℙ

⁢

{

A

i

}

⁢

ℙ

⁢

{

E

|

A

i

}

=

ℙ

⁢

{

E

∩

B

}

∑

i

ℙ

⁢

{

E

∩

A

i

}

=

=

ℙ

⁢

{

E

∩

B

}

ℙ

⁢

{

E

}

=

ℙ

{

B

|

E

}

.

Title

proof of Bayes’ Theorem

Canonical name

ProofOfBayesTheorem

Date of creation

2013-03-22 12:45:09

Last modified on

2013-03-22 12:45:09

Owner

ariels (338)

Last modified by

ariels (338)

Numerical id

5

Author

ariels (338)

Entry type

Proof

Classification

msc 60-00

Classification

msc 62A01

Generated on Sat Feb 10 11:52:07 2018 by

LaTeXML
properties of expected value

properties of expected value

1)

(normalization)

Let

X

be almost surely

constant random
variable

, i.e.

Pr

⁡

{

X

=

c

}

=

1

; then

E

⁢

[

X

]

=

c

.

2)

(linearity)

Let

X

,

Y

be

random variables

such that

E

⁢

[

|

X

|

]

<

∞

and

E

⁢

[

|

Y

|

]

<

∞

and let

a

,

b

be real numbers; then

E

⁢

[

|

a

⁢

X

+

b

⁢

Y

|

]

<

∞

and

E

⁢

[

a

⁢

X

+

b

⁢

Y

]

=

a

⁢

E

⁢

[

X

]

+

b

⁢

E

⁢

[

Y

]

.

3)

(monotonicity)

Let

X

,

Y

be random variables such that

Pr

⁡

{

X

≤

Y

}

=

1

and

E

⁢

[

|

X

|

]

<

∞

,

E

⁢

[

|

Y

|

]

<

∞

; then

E

⁢

[

X

]

≤

E

⁢

[

Y

]

.

Proof.

1) Let’s define

F

=

{

ω

∈

Ω

:

X

⁢

(

ω

)

=

c

}

;

Then by

hypothesis

Pr

⁡

{

Ω

\

F

}

=

0

and

Pr

⁡

{

F

}

=

1

.

We have:

E

⁢

[

X

]

=

∫

Ω

X

⁢

(

ω

)

⁢

𝑑

P

=

∫

Ω

\

F

X

⁢

(

ω

)

⁢

𝑑

P

+

∫

F

X

⁢

(

ω

)

⁢

𝑑

P

=

∫

F

X

⁢

(

ω

)

⁢

𝑑

P

=

∫

F

c

⁢

𝑑

P

=

c

⁢

Pr

⁡

{

F

}

=

c

.

2)

[to be done]

.

3) Let’s define

F

=

{

ω

∈

Ω

:

X

⁢

(

ω

)

≤

Y

⁢

(

ω

)

}

;

Then by hypothesis

Pr

⁡

{

Ω

\

F

}

=

0

and

Pr

⁡

{

F

}

=

1

.

We have, keeping in mind property 2),

E

⁢

[

Y

]

-

E

⁢

[

X

]

=

E

⁢

[

Y

-

X

]

=

∫

Ω

[

Y

⁢

(

ω

)

-

X

⁢

(

ω

)

]

⁢

𝑑

P

=

∫

Ω

\

F

[

Y

⁢

(

ω

)

-

X

⁢

(

ω

)

]

⁢

𝑑

P

+

∫

F

[

Y

⁢

(

ω

)

-

X

⁢

(

ω

)

]

⁢

𝑑

P

=

∫

F

[

Y

⁢

(

ω

)

-

X

⁢

(

ω

)

]

⁢

𝑑

P

≥

0

.

∎

Title

properties of expected value

Canonical name

PropertiesOfExpectedValue

Date of creation

2013-03-22 16:16:05

Last modified on

2013-03-22 16:16:05

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

6

Author

Andrea Ambrosio (7332)

Entry type

Theorem

Classification

msc 60-00

Generated on Sat Feb 10 11:52:09 2018 by

LaTeXML
uniform (continuous) random variable

uniform (continuous) random variable

A

random variable

X

is said to be a

() random variable

with parameters

a

and

b

if its

probability density function

is given by

f

X

⁢

(

x

)

=

1

b

-

a

,

x

∈

[

a

,

b

]

,

and is denoted

X

∼

U

⁢

(

a

,

b

)

.

Notes:

1.

They are also called

rectangular distributions

, considers that all points in the interval

[

a

,

b

]

have the same mass.

2.

E

⁢

[

X

]

=

a

+

b

2

3.

V

⁢

a

⁢

r

⁢

[

X

]

=

(

b

-

a

)

2

12

4.

M

X

⁢

(

t

)

=

e

b

⁢

t

-

e

a

⁢

t

(

b

-

a

)

⁢

t

Title

uniform (continuous) random variable

Canonical name

UniformcontinuousRandomVariable

Date of creation

2013-03-22 11:54:18

Last modified on

2013-03-22 11:54:18

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

10

Author

mathcam (2727)

Entry type

Definition

Classification

msc 60-00

Synonym

uniform random variable

Synonym

rectangular distribution

Synonym

uniform distribution

Generated on Sat Feb 10 11:52:12 2018 by

LaTeXML
uniform (discrete) random variable

uniform (discrete) random variable

X

is a

uniform (discrete)

random variable

with parameter

N

if

f

X

⁢

(

x

)

=

1

N

,

x

=

{

1

,

2

,

…

,

N

}

Parameters:

⋆

N

∈

{

1

,

2

,

…

}

Syntax:

X

∼

U

⁢

{

N

}

Notes:

1.

X

represents the experiment in which all N outcomes are equally likely to ocurr.

2.

E

⁢

[

X

]

=

N

+

1

2

3.

V

⁢

a

⁢

r

⁢

[

X

]

=

N

2

-

1

12

4.

M

X

⁢

(

t

)

=

∑

j

=

1

N

1

N

⁢

e

j

⁢

t

Title

uniform (discrete) random variable

Canonical name

UniformdiscreteRandomVariable

Date of creation

2013-03-22 11:54:09

Last modified on

2013-03-22 11:54:09

Owner

Riemann (23)

Last modified by

Riemann (23)

Numerical id

7

Author

Riemann (23)

Entry type

Definition

Classification

msc 60-00

Classification

msc 92B05

Classification

msc 92-01

Synonym

uniform random variable

Synonym

discrete uniform distribution

Generated on Sat Feb 10 11:52:14 2018 by

LaTeXML
conditional independence

conditional independence

Let

(

Ω

,

ℱ

,

P

)

be a

probability space

.

Conditional Independence Given an Event

Given an event

C

∈

ℱ

:

1.

Two events

A

and

B

in

ℱ

are said to be

conditionally independent

given

C

if we have the following equality of

conditional probabilities

:

P

(

A

∩

B

|

C

)

=

P

(

A

|

C

)

P

(

B

|

C

)

.

2.

Two sub

sigma algebras

ℱ

1

,

ℱ

2

of

ℱ

are

conditionally independent given

C

if any two events

A

∈

ℱ

1

and

B

∈

ℱ

2

are conditionally independent given

C

.

3.

Two real

random variables

X

,

Y

:

Ω

→

ℝ

are

conditionally independent given event

C

if

ℱ

X

and

ℱ

Y

, the sub sigma algebras generated by (

http://planetmath.org/MathcalFMeasurableFunction

)

X

and

Y

are conditionally independent given

C

.

Conditional Independence Given a Sigma Algebra

Given a sub sigma algebra

𝒢

of

ℱ

:

1.

Two events

A

and

B

in

ℱ

are said to be

conditionally independent given

G

if we have the following equality of conditional probabilities (as random variables) (

http://planetmath.org/ProbabilityConditioningOnASigmaAlgebra

):

P

(

A

∩

B

|

𝒢

)

=

P

(

A

|

𝒢

)

P

(

B

|

𝒢

)

.

2.

Two sub sigma algebras

ℱ

1

,

ℱ

2

of

ℱ

are

conditionally independent given

G

if any two events

A

∈

ℱ

1

and

B

∈

ℱ

2

are conditionally independent given

𝒢

.

3.

Two real random variables

X

,

Y

:

Ω

→

ℝ

are

conditionally independent given event

G

if

ℱ

X

and

ℱ

Y

, the sub sigma algebras generated by

X

and

Y

are conditionally independent given

𝒢

.

4.

Finally, we can define

conditional

idependence given a random variable

, say

Z

:

Ω

→

ℝ

in each of the above three items by setting

𝒢

=

ℱ

Z

.

Title

conditional independence

Canonical name

ConditionalIndependence

Date of creation

2013-03-22 16:25:09

Last modified on

2013-03-22 16:25:09

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

4

Author

CWoo (3771)

Entry type

Definition

Classification

msc 60A05

Defines

conditionally independent

Generated on Sat Feb 10 11:52:23 2018 by

LaTeXML
example of pairwise independent events that are not totally independent

example of pairwise independent events that are not totally independent

Consider a fair tetrahedral die whose sides are painted red, green, blue, and white. Roll the die. Let

X

r

,

X

g

,

X

b

be the events that die falls on a side that have red, green, and blue color

components

, respectively. Then

P

⁢

(

X

r

)

=

P

⁢

(

X

g

)

=

P

⁢

(

X

b

)

=

1

2

,

P

⁢

(

X

r

∩

X

g

)

=

P

⁢

(

X

w

)

=

1

4

=

P

⁢

(

X

r

)

⁢

P

⁢

(

X

g

)

,

but

P

⁢

(

X

r

∩

X

g

∩

X

b

)

=

1

4

≠

1

8

=

P

⁢

(

X

r

)

⁢

P

⁢

(

X

g

)

⁢

P

⁢

(

X

b

)

.

Title

example of pairwise independent events that are not totally

independent

Canonical name

ExampleOfPairwiseIndependentEventsThatAreNotTotallyIndependent

Date of creation

2013-03-22 13:38:51

Last modified on

2013-03-22 13:38:51

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

6

Author

bbukh (348)

Entry type

Example

Classification

msc 60A05

Generated on Sat Feb 10 11:52:25 2018 by

LaTeXML
independent

independent

In a

probability space

, we say that the

random events

A

1

,

…

,

A

n

are

independent

if

P

⁢

(

A

i

1

∩

A

i

2

∩

…

∩

A

i

k

)

=

P

⁢

(

A

i

1

)

⁢

…

⁢

P

⁢

(

A

i

k

)

for all

i

1

,

…

,

i

k

such that

1

≤

i

1

<

i

2

<

⋯

<

i

k

≤

n

.

An arbitrary family of random events is independent if every finite subfamily is independent.

The

random variables

X

1

,

…

,

X

n

are independent if, given any Borel sets

B

1

,

…

,

B

n

, the random events

[

X

1

∈

B

1

]

,

…

,

[

X

n

∈

B

n

]

are independent. This is

equivalent

to saying that

F

X

1

,

…

,

X

n

=

F

X

1

⁢

…

⁢

F

X

n

where

F

X

1

,

…

,

F

X

n

are the

distribution functions

of

X

1

,

…

,

X

n

, respectively, and

F

X

1

,

…

,

X

n

is the

joint distribution function

. When the

density functions

f

X

1

,

…

,

f

X

n

and

f

X

1

,

…

,

X

n

exist, an equivalent condition for independence is that

f

X

1

,

…

,

X

n

=

f

X

1

⁢

…

⁢

f

X

n

.

An arbitrary family of random variables is independent if every finite subfamily is independent.

Title

independent

Canonical

name

Independent

Date of creation

2013-03-22 12:02:15

Last modified on

2013-03-22 12:02:15

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

11

Author

Koro (127)

Entry type

Definition

Classification

msc 60A05

Generated on Sat Feb 10 11:52:27 2018 by

LaTeXML
independent sigma algebras

independent sigma algebras

Let

(

Ω

,

ℬ

,

P

)

be a

probability space

. Let

ℬ

1

and

ℬ

2

be two sub

sigma algebras

of

ℬ

. Then

ℬ

1

and

ℬ

2

are said to be

if for any pair of events

B

1

∈

ℬ

1

and

B

2

∈

ℬ

2

:

P

⁢

(

B

1

∩

B

2

)

=

P

⁢

(

B

1

)

⁢

P

⁢

(

B

2

)

.

More generally, a

finite set

of sub-

σ

-

algebras

ℬ

1

,

…

,

ℬ

n

is

independent

if for any set of events

B

i

∈

ℬ

i

,

i

=

1

,

…

,

n

:

P

⁢

(

B

1

∩

⋯

∩

B

n

)

=

P

⁢

(

B

1

)

⁢

⋯

⁢

P

⁢

(

B

n

)

.

An arbitrary set

𝒮

of sub-

σ

-algebras is

mutually independent

if any finite subset of

𝒮

is independent.

The above definitions are

generalizations

of the notions of independence (

http://planetmath.org/Independent

) for events and for

random variables

:

1.

Events

B

1

,

…

,

B

n

(in

Ω

) are

mutually independent

if the sigma algebras

σ

⁢

(

B

i

)

:=

{

∅

,

B

i

,

Ω

-

B

i

,

Ω

}

are mutually independent.

2.

Random variables

X

1

,

…

,

X

n

defined on

Ω

are

mutually independent

if the sigma algebras

ℬ

X

i

generated by (

http://planetmath.org/MathcalFMeasurableFunction

) the

X

i

’s are mutually independent.

In general, mutual independence among events

B

i

, random variables

X

j

, and sigma algebras

ℬ

k

means the mutual independence among

σ

⁢

(

B

i

)

,

ℬ

X

j

, and

ℬ

k

.

Remark

. Even when random variables

X

1

,

…

,

X

n

are defined on different probability spaces

(

Ω

i

,

ℬ

i

,

P

i

)

, we may form the

product

(

http://planetmath.org/InfiniteProductMeasure

) of these spaces

(

Ω

,

ℬ

,

P

)

so that

X

i

(by abuse of notation) are now defined on

Ω

and their independence can be discussed.

Title

independent sigma algebras

Canonical name

IndependentSigmaAlgebras

Date of creation

2013-03-22 16:22:58

Last modified on

2013-03-22 16:22:58

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 60A05

Synonym

mutually independent

σ

-algebras

Defines

mutually independent sigma algebras

Generated on Sat Feb 10 11:52:36 2018 by

LaTeXML
operations with events

operations with events

Two events E and F are

disjoint

or mutually exclusive iff their

intersection

is empty:

E

∩

F

=

E

⁢

F

=

∅

Title

operations with events

Canonical name

OperationsWithEvents

Date of creation

2013-03-22 18:43:20

Last modified on

2013-03-22 18:43:20

Owner

polarbear (3475)

Last modified by

polarbear (3475)

Numerical id

4

Author

polarbear (3475)

Entry type

Derivation

Classification

msc 60A05

Defines

mutually exclusive events

Generated on Sat Feb 10 11:52:38 2018 by

LaTeXML
random event

random event

In a

probability space

,

measurable sets

are usually called (random) events.

Title

random event

Canonical name

RandomEvent

Date of creation

2013-03-22 13:14:01

Last modified on

2013-03-22 13:14:01

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

4

Author

Koro (127)

Entry type

Definition

Classification

msc 60A05

Synonym

event

Generated on Sat Feb 10 11:52:39 2018 by

LaTeXML
almost everywhere

almost everywhere

Let

(

X

,

𝔅

,

μ

)

be a

measure space

. A condition holds

almost everywhere

on

X

if it holds “with

probability

1

,” i.e. if it holds everywhere except for a subset of

X

with

measure

0

. For example, let

f

and

g

be nonnegative

functions

on

X

. Suppose we want a

sufficient

condition on functions

f

⁢

(

x

)

and

g

⁢

(

x

)

such that the

relation

∫

X

f

⁢

𝑑

μ

⁢

(

x

)

≤

∫

X

g

⁢

𝑑

μ

⁢

(

x

)

(1)

holds. Certainly

f

⁢

(

x

)

≤

g

⁢

(

x

)

for all

x

∈

X

is a sufficient condition, but in fact it’s enough to have

f

⁢

(

x

)

≤

g

⁢

(

x

)

almost surely on

X

. In fact, we can loosen the above non-negativity condition to only require that

f

and

g

are almost surely nonnegative as well.

If

X

=

[

0

,

1

]

, then

g

might be less than

f

on the

Cantor set

, an

uncountable set

with measure

0

, and still

satisfy

the condition. We say that

f

≤

g

almost everywhere (often abbreviated

a.e.

).

Note that this is the of the “almost surely” from probabilistic measure .

Title

almost everywhere

Canonical name

AlmostEverywhere

Date of creation

2013-03-22 12:20:58

Last modified on

2013-03-22 12:20:58

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 60A10

Synonym

almost surely

Synonym

a.s.

Synonym

a.e.

Synonym

almost all

Generated on Sat Feb 10 11:52:41 2018 by

LaTeXML