aarjaneiro/planetmath huggingface dataset:
Bonferroni inequalities

Bonferroni inequalities

Let

E

⁢

(

1

)

,

E

⁢

(

2

)

,

…

,

E

⁢

(

n

)

be events in a

sample space

. Define

S

1

:=

∑

i

=

1

n

Pr

⁡

(

E

⁢

(

i

)

)

S

2

:=

∑

i

<

j

Pr

⁡

(

E

⁢

(

i

)

∩

E

⁢

(

j

)

)

,

and for

2

<

k

≤

n

,

S

k

:=

∑

Pr

⁡

(

E

⁢

(

i

1

)

∩

⋯

∩

E

⁢

(

i

k

)

)

where the summation is taken over all ordered

k

-tuples of distinct integers.

Theorem

For odd

k

,

1

≤

k

≤

n

,

Pr

⁡

(

E

⁢

(

1

)

∪

⋯

∪

E

⁢

(

n

)

)

≤

∑

j

=

1

k

(

-

1

)

j

+

1

⁢

S

j

,

and for even

k

,

2

≤

k

≤

n

,

Pr

⁡

(

E

⁢

(

1

)

∪

⋯

∪

E

⁢

(

n

)

)

≥

∑

j

=

1

k

(

-

1

)

j

+

1

⁢

S

j

,

Remark

When

k

=

1

, the

Bonferroni inequality

is also known as the union bound.
When

k

=

n

, we have an equality, also known as the

inclusion-exclusion principle

.

Title

Bonferroni inequalities

Canonical name

BonferroniInequalities

Date of creation

2013-03-22 14:30:40

Last modified on

2013-03-22 14:30:40

Owner

kshum (5987)

Last modified by

kshum (5987)

Numerical id

9

Author

kshum (5987)

Entry type

Theorem

Classification

msc 60A99

Related topic

BrunsPureSieve

Defines

union bound

Generated on Sat Feb 10 11:53:29 2018 by

LaTeXML
Boole inequality

Boole inequality

Let

(

Ω

,

ℱ

,

P

)

be a

probability space

and

{

A

1

,

A

2

,

⋯

}

be a

sequence

of events
such that

{

A

1

,

A

2

,

⋯

}

⊂

ℱ

. Then

P

⁢

(

⋃

n

=

1

∞

A

n

)

≤

∑

n

=

1

∞

P

⁢

(

A

n

)

.

Title

Boole inequality

Canonical name

BooleInequality

Date of creation

2013-03-22 15:45:07

Last modified on

2013-03-22 15:45:07

Owner

georgiosl (7242)

Last modified by

georgiosl (7242)

Numerical id

4

Author

georgiosl (7242)

Entry type

Theorem

Classification

msc 60A99

Generated on Sat Feb 10 11:53:31 2018 by

LaTeXML
Boole inequality, proof of

Boole inequality, proof of

Let

{

B

1

,

B

2

,

⋯

}

be a

sequence

defined by:

B

i

=

A

i

∖

⋃

k

=

1

i

-

1

A

k

Clearly

B

i

∈

ℱ

,

∀

i

∈

ℕ

, since

ℱ

is

σ

-algebra, they are a

disjoint

family and :

⋃

n

=

1

i

A

n

=

⋃

n

=

1

i

B

n

,

∀

i

∈

ℕ

and since

P

is a

measure

over

ℱ

it follows that :

P

⁢

(

⋃

n

=

1

i

B

n

)

=

∑

n

=

1

i

P

⁢

(

B

n

)

,

∀

i

∈

ℕ

Clearly

B

i

⊂

A

i

, then

P

⁢

(

B

i

)

≤

P

⁢

(

A

i

)

because measures are

http://planetmath.org/node/4460

monotonic, then it follows that :

P

⁢

(

⋃

n

=

1

i

B

n

)

≤

∑

n

=

1

i

P

⁢

(

A

n

)

,

∀

i

∈

ℕ

finally taking

n

→

∞

:

P

⁢

(

⋃

n

=

1

∞

A

n

)

=

P

⁢

(

⋃

n

=

1

∞

B

n

)

≤

∑

n

=

1

∞

P

⁢

(

A

n

)

the latter is valid because the measure continuity , and is the proof of the

theorem

Title

Boole inequality

, proof of

Canonical name

BooleInequalityProofOf

Date of creation

2013-03-22 15:47:18

Last modified on

2013-03-22 15:47:18

Owner

Bunder (13010)

Last modified by

Bunder (13010)

Numerical id

6

Author

Bunder (13010)

Entry type

Proof

Classification

msc 60A99

Generated on Sat Feb 10 11:53:33 2018 by

LaTeXML
Borel-Cantelli lemma

Borel-Cantelli lemma

Let

A

1

,

A

2

,

…

be

random events

in a

probability space

.

1.

If

∑

n

=

1

∞

P

⁢

(

A

n

)

<

∞

, then

P

⁢

(

A

n

⁢

i

.

o

.

)

=

0

;

2.

If

A

1

,

A

2

,

…

are

independent

, and

∑

n

=

1

∞

P

⁢

(

A

n

)

=

∞

,
then

P

⁢

(

A

n

⁢

i

.

o

.

)

=

1

where

A

=

[

A

n

⁢

i

.

o

.

]

represents the event “

A

n

happens for infinitely many values of

n

.”
Formally,

A

=

lim sup

⁡

A

n

, which is a

limit superior of sets

.

Title

Borel-Cantelli lemma

Canonical name

BorelCantelliLemma

Date of creation

2013-03-22 13:13:18

Last modified on

2013-03-22 13:13:18

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

7

Author

Koro (127)

Entry type

Theorem

Classification

msc 60A99

Generated on Sat Feb 10 11:53:34 2018 by

LaTeXML
Chebyshev’s inequality

Chebyshev’s inequality

Let

X

∈

𝐋

2

be a real-valued

random variable

with mean

μ

=

𝔼

⁢

[

X

]

and

variance

σ

2

=

Var

⁡

[

X

]

. Then for any standard of accuracy

t

>

0

,

ℙ

{

|

X

-

μ

|

≥

t

}

≤

σ

2

t

2

.

Note:

There is another Chebyshev’s inequality (

http://planetmath.org/ChebyshevsInequality

), which is unrelated.

Title

Chebyshev’s inequality

Canonical name

ChebyshevsInequality

Date of creation

2013-03-22 12:47:55

Last modified on

2013-03-22 12:47:55

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

6

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 60A99

Related topic

MarkovsInequality

Related topic

ChebyshevsInequality

Generated on Sat Feb 10 11:53:36 2018 by

LaTeXML
conditional probability

conditional probability

Let

(

Ω

,

𝔅

,

μ

)

be a

probability space

, and let

X

,

Y

∈

𝔅

be events.

The

conditional probability

of

X

given

Y

is defined as

μ

(

X

|

Y

)

=

μ

⁢

(

X

∩

Y

)

μ

⁢

(

Y

)

(1)

provided

μ

⁢

(

Y

)

>

0

.
(If

μ

⁢

(

Y

)

=

0

, then

μ

(

X

|

Y

)

is not defined.)

If

μ

⁢

(

X

)

>

0

and

μ

⁢

(

Y

)

>

0

, then

μ

(

X

|

Y

)

μ

(

Y

)

=

μ

(

X

∩

Y

)

=

μ

(

Y

|

X

)

μ

(

X

)

,

(2)

and so also

μ

(

X

|

Y

)

=

μ

(

Y

|

X

)

μ

(

X

)

μ

⁢

(

Y

)

,

(3)

which is Bayes’ Theorem.

Title

conditional probability

Canonical name

ConditionalProbability

Date of creation

2013-03-22 12:21:54

Last modified on

2013-03-22 12:21:54

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

8

Author

yark (2760)

Entry type

Definition

Classification

msc 60A99

Related topic

ConditionalEntropy

Related topic

BayesTheorem

Related topic

ConditionalExpectation

Generated on Sat Feb 10 11:53:38 2018 by

LaTeXML
corollary of Borel-Cantelli lemma

corollary of Borel-Cantelli lemma

Under independence, we have an equivalence.

sketch: You have only to think about the counterpositive of the lemmas and use the Kolmogorov 0-1 law or try to do it by

contradiction

which is simplier.

Title

corollary of Borel-Cantelli lemma

Canonical

name

CorollaryOfBorelCantelliLemma

Date of creation

2013-03-22 15:22:09

Last modified on

2013-03-22 15:22:09

Owner

renato (9974)

Last modified by

renato (9974)

Numerical id

10

Author

renato (9974)

Entry type

Corollary

Classification

msc 60A99

Generated on Sat Feb 10 11:53:40 2018 by

LaTeXML
criterion for almost-sure convergence

criterion for almost-sure convergence

Let

X

1

,

X

2

,

…

and

X

be

random variables

.
If, for every

ϵ

>

0

, the sum

∑

n

=

1

∞

ℙ

(

|

X

n

-

X

|

>

ϵ

)

is finite,
then

X

n

converge

to

X

almost surely.

Proof.

By the

Borel-Cantelli lemma

, we have

ℙ

(

lim sup

n

{

|

X

n

-

X

|

>

ϵ

}

)

=

0

.
But

lim sup

n

{

|

X

n

-

X

|

>

ϵ

}

is the same as the event

{

lim sup

n

⁡

|

X

n

-

X

|

>

ϵ

}

.
(The latter event involves the

limit superior

of

numbers

(

http://planetmath.org/LimitSuperior

); the former involves the
limit superior of

sets

(

http://planetmath.org/InfinitelyOften

).)
So taking the limit

ϵ

↘

0

,
we have

ℙ

(

lim sup

n

|

X

n

-

X

|

>

0

)

=

0

,
or equivalently

ℙ

(

lim sup

n

|

X

n

-

X

|

=

0

)

=

1

.
∎

Title

criterion for almost-sure convergence

Canonical name

CriterionForAlmostsureConvergence

Date of creation

2013-03-22 15:54:45

Last modified on

2013-03-22 15:54:45

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

15

Author

stevecheng (10074)

Entry type

Corollary

Classification

msc 60A99

Synonym

corollary of Borel-Cantelli lemma

Generated on Sat Feb 10 11:53:42 2018 by

LaTeXML
cumulative distribution function

cumulative distribution function

Let

X

be a

random variable

. Define

F

X

:

R

→

[

0

,

1

]

as

F

X

⁢

(

x

)

=

Pr

⁡

[

X

≤

x

]

for all

x

. The

function

F

X

⁢

(

x

)

is called the

cumulative distribution function

of

X

.

Every cumulative distribution function satisfies the following

properties

:

1.

lim

x

→

-

∞

⁡

F

X

⁢

(

x

)

=

0

and

lim

x

→

+

∞

⁡

F

X

⁢

(

x

)

=

1

,

2.

F

X

is a monotonically nondecreasing function,

3.

F

X

is

continuous

from the right,

4.

Pr

⁡

[

a

<

X

≤

b

]

=

F

X

⁢

(

b

)

-

F

X

⁢

(

a

)

.

If

X

is a discrete random variable, then the cumulative

distribution

can be expressed as

F

X

⁢

(

x

)

=

∑

k

≤

x

Pr

⁡

[

X

=

k

]

.

Similarly, if

X

is a continuous random variable, then

F

X

⁢

(

x

)

=

∫

-

∞

x

f

X

⁢

(

y

)

⁢

𝑑

y

where

f

X

is the

density

distribution function

.

Title

cumulative distribution function

Canonical

name

CumulativeDistributionFunction

Date of creation

2013-03-22 11:53:38

Last modified on

2013-03-22 11:53:38

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

10

Author

bbukh (348)

Entry type

Definition

Classification

msc 60A99

Classification

msc 46L05

Classification

msc 82-00

Classification

msc 83-00

Classification

msc 81-00

Related topic

DistributionFunction

Related topic

DensityFunction

Generated on Sat Feb 10 11:53:44 2018 by

LaTeXML
Markov’s inequality

Markov’s inequality

For a non-negative

random variable

X

and a standard of accuracy

d

>

0

, Markov’s inequality that

P

[

X

≥

d

]

≤

1

d

E

[

X

]

.

Title

Markov’s inequality

Canonical name

MarkovsInequality

Date of creation

2013-03-22 12:34:08

Last modified on

2013-03-22 12:34:08

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

6

Author

Andrea Ambrosio (7332)

Entry type

Theorem

Classification

msc 60A99

Synonym

Markoff’s inequality

Related topic

ChebyshevsInequality2

Related topic

KolmogorovsInequality

Related topic

KolmogorovsMartingaleInequality

Related topic

ReverseMarkovInequality

Generated on Sat Feb 10 11:53:45 2018 by

LaTeXML
median of a distribution

median of a distribution

Given a probability distribution (

density

)

function

f

X

⁢

(

x

)

on

Ω

over a

random variable

X

, with the associated

probability measure

P

, a

median

m

of

f

X

is a real number such that

1.

P

(

X

≤

m

)

≥

1

2

,

2.

P

(

X

≥

m

)

≥

1

2

.

The median is also known as the

50

th

-

percentile

or the

second quartile

.

Examples:

•

An example from a

discrete distribution

. Let

Ω

=

ℝ

. Suppose the random variable

X

has the following

distribution

:

P

(

X

=

0

)

=

0.99

and

P

(

X

=

1000

)

=

0.01

. Then we can easily see the median is 0.

•

Another example from a discrete distribution. Again, let

Ω

=

ℝ

. Suppose the random variable

X

has distribution

P

(

X

=

0

)

=

0.5

and

P

(

X

=

1000

)

=

0.5

. Then we see that the median is not unique. In fact, all real values in the interval

[

0

,

1000

]

are medians.

•

In practice, however, the median may be calculated as follows: if there are

N

numeric data points, then by

ordering

the data values (either non-decreasingly or non-increasingly),

(a)

the

(

N

+

1

2

)

-th data point is the median if

N

is odd, and

(b)

the midpoint of the

(

N

-

1

)

th and the

(

N

+

1

)

th data points is the median if

N

is even.

•

The median of a

normal distribution

(with mean

μ

and

variance

σ

2

) is

μ

. In fact, for a normal distribution, mean = median = mode.

•

The median of a

uniform distribution

in the interval

[

a

,

b

]

is

(

a

+

b

)

/

2

.

•

The median of a

Cauchy distribution

with

location parameter

t and scale parameter s is the location parameter.

•

The median of an

exponential distribution

with location parameter

μ

and scale parameter

β

is the scale parameter times the natural log of 2,

β

⁢

ln

⁡

2

.

•

The median of a

Weibull distribution

with shape parameter

γ

, location parameter

μ

, and scale parameter

α

is

α

⁢

(

ln

⁡

2

)

1

/

γ

+

μ

.

Title

median of a distribution

Canonical name

MedianOfADistribution

Date of creation

2013-03-22 14:24:10

Last modified on

2013-03-22 14:24:10

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Definition

Classification

msc 60A99

Classification

msc 62-07

Synonym

second quartile

Defines

median

Generated on Sat Feb 10 11:53:48 2018 by

LaTeXML
mode

mode

Given a probability distribution (

density

)

function

f

X

⁢

(

x

)

with

random variable

X

and

x

∈

ℝ

, a

mode

of

f

X

⁢

(

x

)

is a real number

α

such that:

1.

f

X

⁢

(

α

)

≠

min

⁡

(

f

X

⁢

(

x

)

)

,

2.

f

X

⁢

(

α

)

≥

f

X

⁢

(

z

)

for all

z

∈

ℝ

.

The

mode of

f

X

is the set of all modes of

f

X

(It is also customary to say denote the mode of

f

X

to be

elements

within the mode of

f

X

). If the mode contains one element, then we say that

f

X

is

unimodal

. If it has two elements, then

f

X

is called

bimodal

. When

f

X

has more than two modes, it is called

multimodal

.

•

if

Ω

=

{

0

,

1

,

2

,

2

,

3

,

4

,

4

,

4

,

5

,

5

,

6

,

7

,

8

}

is the

sample space

for the random variable

X

, then the mode of the

distribution function

f

X

is 4.

•

if

Ω

=

{

0

,

2

,

4

,

5

,

6

,

6

,

7

,

9

,

11

,

11

,

14

,

18

}

is the sample space for

X

, then the modes of

f

X

are 6 and 11 and

f

X

is bimodal.

•

For a

binomial distribution

with mean

n

⁢

p

and

variance

n

⁢

p

⁢

(

1

-

p

)

, the mode is

{

α

∣

p

⁢

(

n

+

1

)

-

1

≤

α

≤

p

⁢

(

n

+

1

)

}

.

•

For a

Poisson distribution

with integral sample space and mean

λ

, if

λ

is non-integral, then the mode is the largest

integer

less than or equal to

λ

; if

λ

is an integer, then both

λ

and

λ

-

1

are modes.

•

For a

normal distribution

with mean

μ

and

standard deviation

σ

, the mode is

μ

.

•

For a

gamma distribution

with the shape parameter

γ

,

location parameter

μ

, and scale parameter

β

, the mode is

γ

-

1

if

γ

>

1

.

•

Both the Pareto and the

exponential distributions

have mode = 0.

Title

mode

Canonical name

Mode

Date of creation

2013-03-22 14:23:33

Last modified on

2013-03-22 14:23:33

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

4

Author

CWoo (3771)

Entry type

Definition

Classification

msc 60A99

Generated on Sat Feb 10 11:53:50 2018 by

LaTeXML
Multidimensional Chebyshev’s inequality

Multidimensional Chebyshev’s inequality

Let

X

be an N-dimensional

random variable

with mean

μ

=

𝔼

⁢

[

X

]

and

covariance matrix

V

=

𝔼

⁢

[

(

X

-

μ

)

⁢

(

X

-

μ

)

T

]

.

If

V

is

invertible

(i.e., strictly

positive

), for any

t

>

0

:

Pr

⁡

(

(

X

-

μ

)

T

⁢

V

-

1

⁢

(

X

-

μ

)

>

t

)

≤

N

t

2

Proof:

V

is positive, so

V

-

1

is.
Define the random variable

y

=

(

X

-

μ

)

T

⁢

V

-

1

⁢

(

X

-

μ

)

y

is positive, then Markov’s inequality holds:

Pr

⁡

(

(

X

-

μ

)

T

⁢

V

-

1

⁢

(

X

-

μ

)

>

t

)

=

Pr

⁡

(

y

>

t

)

=

Pr

⁡

(

y

>

t

2

)

≤

𝔼

⁢

[

y

]

t

2

Since

V

is

symmetric

, a

rotation

R

(i.e.,

R

⁢

R

T

=

R

T

⁢

R

=

I

) and a

diagonal matrix

D

(i.e.,

i

≠

j

⇒

D

i

,

j

=

0

) exist such that

V

=

R

T

⁢

D

⁢

R

Since

V

is positive

D

i

⁢

i

>

0

.
Besides

V

-

1

=

R

-

1

⁢

D

-

1

⁢

(

R

T

)

-

1

=

R

T

⁢

D

-

1

⁢

R

clearly

[

D

-

1

]

i

⁢

i

=

1

D

i

⁢

i

.

Define

Z

=

R

⁢

(

X

-

μ

)

.

The following identities hold:

𝔼

⁢

[

Z

⁢

Z

T

]

=

R

⁢

𝔼

⁢

[

(

X

-

μ

)

⁢

(

X

-

μ

)

T

]

⁢

R

T

=

R

⁢

R

T

⁢

D

⁢

R

⁢

R

T

=

D

⇒

∀

i

𝔼

⁢

[

Z

i

2

]

=

D

i

⁢

i

and

y

=

Z

T

⁢

R

⁢

V

-

1

⁢

R

T

⁢

Z

=

Z

T

⁢

D

-

1

⁢

Z

=

∑

i

=

1

N

Z

i

2

D

i

⁢

i

then

𝔼

⁢

[

y

]

=

∑

i

=

1

N

𝔼

⁢

[

Z

i

2

]

D

i

⁢

i

=

N

Title

Multidimensional Chebyshev’s inequality

Canonical name

MultidimensionalChebyshevsInequality

Date of creation

2013-03-22 18:17:55

Last modified on

2013-03-22 18:17:55

Owner

daniWk (21206)

Last modified by

daniWk (21206)

Numerical id

5

Author

daniWk (21206)

Entry type

Theorem

Classification

msc 60A99

Generated on Sat Feb 10 11:53:52 2018 by

LaTeXML
probability conditioning on a sigma algebra

probability conditioning on a sigma algebra

Let

(

Ω

,

𝔅

,

μ

)

be a

probability space

and

B

∈

𝔅

an event. Let

𝔇

be a sub

sigma algebra

of

𝔅

. The

of

B

given

D

is defined to be the

conditional expectation

of the

random variable

1

B

defined on

Ω

, given

𝔇

. We denote this

conditional probability

by

μ

(

B

|

𝔇

)

:=

E

(

1

B

|

𝔇

)

.

1

B

is also known as the

indicator function

.

Similarly, we can define a conditional probability given a random variable. Let

X

be a random variable defined on

Ω

. The

conditional probability of

B

given

X

is defined to be

μ

(

B

|

𝔅

X

)

, where

𝔅

X

is the sub sigma algebra of

𝔅

, generated by (

http://planetmath.org/MathcalFMeasurableFunction

)

X

. The conditional probability of

B

given

X

is simply written

μ

(

B

|

X

)

.

Remark

. Both

μ

(

B

|

𝔇

)

and

μ

(

B

|

X

)

are random variables, the former is

𝔇

-

measurable

, and the latter is

𝔅

X

-measurable.

Title

probability

conditioning on a sigma algebra

Canonical name

ProbabilityConditioningOnASigmaAlgebra

Date of creation

2013-03-22 16:25:05

Last modified on

2013-03-22 16:25:05

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 60A99

Classification

msc 60A10

Generated on Sat Feb 10 11:54:01 2018 by

LaTeXML
proof of Borel-Cantelli 1

proof of Borel-Cantelli 1

Let

B

k

be the event

∪

i

=

k

∞

A

i

for

k

=

1

,

2

,

…

,

. If

x

is in the event

A

i

’s i.o., then

x

∈

B

k

for all

k

. So

x

∈

∩

k

=

1

∞

B

k

.

Conversely

, if

x

∈

B

k

for all

k

, then we can show that

x

is in

A

i

’s i.o. Indeed,

x

∈

B

1

=

∪

i

=

1

∞

A

i

means that

x

∈

A

j

⁢

(

1

)

for some

j

⁢

(

1

)

. However

x

∈

B

j

⁢

(

1

)

+

1

implies that

x

∈

A

j

⁢

(

2

)

for some

j

⁢

(

2

)

that is

strictly

larger than

j

⁢

(

1

)

. Thus we can produce an

infinite

sequence

of integer

j

⁢

(

1

)

<

j

⁢

(

2

)

<

j

⁢

(

3

)

<

…

such that

x

∈

A

j

⁢

(

i

)

for all

i

.

Let

E

be the event

{

x

:

x

∈

A

i

⁢

i.o.

}

. We have

E

=

⋂

k

=

1

∞

⋃

i

=

k

∞

A

i

.

From

E

⊆

B

k

for all

k

, it follows that

P

⁢

(

E

)

≤

P

⁢

(

B

k

)

for all

k

. By

union bound

, we know that

P

⁢

(

B

k

)

≤

∑

i

=

k

∞

P

⁢

(

A

i

)

. So

P

⁢

(

B

k

)

→

0

, by the

hypothesis

that

∑

i

=

1

∞

P

⁢

(

A

i

)

is finite. Therefore,

P

⁢

(

E

)

=

0

.

Title

proof of Borel-Cantelli 1

Canonical

name

ProofOfBorelCantelli1

Date of creation

2013-03-22 14:28:16

Last modified on

2013-03-22 14:28:16

Owner

kshum (5987)

Last modified by

kshum (5987)

Numerical id

6

Author

kshum (5987)

Entry type

Proof

Classification

msc 60A99

Generated on Sat Feb 10 11:54:08 2018 by

LaTeXML
proof of Borel-Cantelli 2

proof of Borel-Cantelli 2

Let

E

denote the set of samples that are in

A

i

infinitely often

. We want to show that the

complement

of

E

has

probability

zero.

As in the

proof of Borel-Cantelli 1

, we know that

E

c

=

⋃

k

=

1

∞

⋂

i

=

k

∞

A

i

c

where the superscript

c

means set complement. But for each

k

,

P

⁢

(

∩

i

=

k

A

i

c

)

=

∏

i

=

k

∞

P

⁢

(

A

i

c

)

=

∏

i

=

k

∞

(

1

-

P

⁢

(

A

i

)

)

Here we use the

assumption

that the event

A

i

’s are

independent

. The inequality

1

-

a

≤

e

-

a

and the assumption that the sum of

P

⁢

(

A

i

)

diverges

together imply that

P

⁢

(

∩

i

=

k

A

i

c

)

≤

exp

⁡

(

-

∑

i

=

k

∞

P

⁢

(

A

i

)

)

=

0

Therefore

E

c

is a union of

countable

number of events, each of them has probability zero. So

P

⁢

(

E

c

)

=

0

.

Title

proof of Borel-Cantelli 2

Canonical name

ProofOfBorelCantelli2

Date of creation

2013-03-22 14:29:35

Last modified on

2013-03-22 14:29:35

Owner

kshum (5987)

Last modified by

kshum (5987)

Numerical id

4

Author

kshum (5987)

Entry type

Proof

Classification

msc 60A99

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

LaTeXML
proof of Chebyshev’s inequality

proof of Chebyshev’s inequality

The proof of Chebyshev’s inequality follows from the application of Markov’s inequality.

Define

Y

=

(

X

-

μ

)

2

. Then

Y

≥

0

is a

random variable

, and

𝔼

⁢

[

Y

]

=

Var

⁡

[

X

]

=

σ

2

.

Applying Markov’s inequality to

Y

, we see that

ℙ

{

|

X

-

μ

|

≥

t

}

=

ℙ

{

Y

≥

t

2

}

≤

1

t

2

𝔼

[

Y

]

=

σ

2

t

2

.

Title

proof of Chebyshev’s inequality

Canonical name

ProofOfChebyshevsInequality

Date of creation

2013-03-22 12:47:58

Last modified on

2013-03-22 12:47:58

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Proof

Classification

msc 60A99

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

LaTeXML
proof of Markov’s inequality

proof of Markov’s inequality

Define

Y

=

{

d

X

≥

d

0

otherwise

.

Then

0

≤

Y

≤

X

. Additionally, it follows immediately from the definition that

Y

is a

random variable

(i.e., that it is

measurable

). Computing the

expected value

of

Y

, we have that

𝔼

[

X

]

≥

𝔼

[

Y

]

=

d

⋅

ℙ

{

X

≥

d

}

,

and the

inequality

follows.

Title

proof of Markov’s inequality

Canonical name

ProofOfMarkovsInequality

Date of creation

2013-03-22 12:47:42

Last modified on

2013-03-22 12:47:42

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

7

Author

Andrea Ambrosio (7332)

Entry type

Proof

Classification

msc 60A99

Generated on Sat Feb 10 11:54:13 2018 by

LaTeXML