aarjaneiro/planetmath huggingface dataset:
wavelet representation of Brownian motion

wavelet representation of Brownian motion

First we define the function

H

⁢

(

t

)

=

{

1

for

⁢

0

≤

t

<

1

2

-

1

for

⁢

1

2

≤

t

≤

1

0

otherwise.

(1)

and the

sequence

of functions

H

n

⁢

(

t

)

=

2

j

/

2

⁢

H

⁢

(

2

j

⁢

t

-

k

)

(2)

for

n

=

2

j

+

k

where

j

>

0

and

0

≤

k

≤

2

j

. We also set

H

0

⁢

(

t

)

=

1

.

Wavelet Representation of Brownian Motion

.

If

{

Z

n

:

0

≤

n

<

∞

}

is a sequence of

independent

Gaussian

random variables

with mean

0

and

variance

1

, then the series defined by

X

t

=

∑

n

=

0

∞

(

Z

n

⁢

∫

0

t

H

n

⁢

(

s

)

⁢

𝑑

s

)

(3)

converges uniformly on

[

0

,

1

]

with

probability

one. Moreover, the process

{

X

t

}

defined by
the limit is a

Brownian motion

for

0

≤

t

≤

1

.

Title

wavelet representation of Brownian motion

Canonical name

WaveletRepresentationOfBrownianMotion

Date of creation

2013-03-22 15:12:51

Last modified on

2013-03-22 15:12:51

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Theorem

Classification

msc 60J65

Synonym

construction of Brownian motion

Generated on Sat Feb 10 12:01:48 2018 by

LaTeXML
memoryless random variable

memoryless random variable

A non-negative-valued

random variable

X

is

memoryless

if

P

(

X

>

s

+

t

∣

X

>

s

)

=

P

(

X

>

t

)

for

s

,

t

≥

0

.

In words, given that a certain event did not occur during time period

s

in the past

, the chance that an event will occur after an additional time period

t

in the future

is the same as the chance that the event would occur after a time period

t

from the beginning, regardless of how long or how short the time period

s

is; the memory is

erased

.

From the definition, we see that

P

(

X

>

t

)

=

P

(

X

>

s

+

t

∣

X

>

s

)

=

P

(

X

>

s

+

t

and

X

>

s

)

P

(

X

>

s

)

=

P

(

X

>

s

+

t

)

P

(

X

>

s

)

,

so

P

(

X

>

s

+

t

)

=

P

(

X

>

s

)

P

(

X

>

t

)

iff

X

is memoryless.

An example of a discrete

memoryless random variable

is the

geometric random variable

, since

P

(

X

>

s

+

t

)

=

(

1

-

p

)

s

+

t

=

(

1

-

p

)

s

(

1

-

p

)

t

=

P

(

X

>

s

)

P

(

X

>

t

)

, where

p

is the

probability

of

X

=success. The

exponential random variable

is an example of a continuous memoryless random variable, which can be proved similarly with

1

-

p

replaced by

e

-

λ

. In fact, the exponential random variable is the only continuous random variable having the memoryless property.

Title

memoryless random variable

Canonical name

MemorylessRandomVariable

Date of creation

2013-03-22 14:39:49

Last modified on

2013-03-22 14:39:49

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 60K05

Classification

msc 60G07

Related topic

MarkovChain

Generated on Sat Feb 10 12:01:50 2018 by

LaTeXML
stopping time

stopping time

Let

(

ℱ

t

)

t

∈

𝕋

be a

filtration

(

http://planetmath.org/FiltrationOfSigmaAlgebras

) on a set

Ω

.
A

random variable

τ

taking values in

𝕋

∪

{

∞

}

is a

stopping time

for the filtration

(

ℱ

t

)

if the event

{

τ

≤

t

}

∈

ℱ

t

for every

t

∈

𝕋

.

Remarks

•

The set

𝕋

is the

index set

for the time variable

t

, and the

σ

-

algebra

ℱ

t

is the

collection

of all events which are observable up to and including time

t

. Then, the condition that

τ

is a stopping time means that the outcome of the event

{

τ

≤

t

}

is known at time

t

.

•

In

discrete

time situations, where

𝕋

=

{

0

,

1

,

2

,

…

}

, the condition that

{

τ

≤

t

}

∈

ℱ

t

is

equivalent

to requiring that

{

τ

=

t

}

∈

ℱ

t

. This is not true for

continuous

time cases where

𝕋

is an

interval

of the

real numbers

and hence

uncountable

, due to the fact that

σ

-algebras are not in general

closed under

taking uncountable unions of events.

•

A random time

τ

is a stopping time for a

stochastic process

(

X

t

)

if it is a stopping time for the

natural filtration

of

X

. That is,

{

τ

≤

t

}

∈

σ

(

X

s

:

s

≤

t

)

.

•

The first time that an

adapted process

X

t

hits a given value or set of values is a stopping time. The

inclusion

of

∞

into the range of

τ

is to cover the case where

X

t

never hits the given values.

•

Stopping time is often used in gambling, when a gambler stops the betting process when he reaches a certain goal. The time it takes to reach this goal is generally not a

deterministic

one. Rather, it is a random variable depending on the current result of the bet, as well as the combined information from all previous bets.

Examples.

A gambler has $1,000 and plays the slot

machine

at $1 per play.

1.

The gambler stops playing when his capital is depleted. The number

τ

=

n

1

of plays that it takes the gambler to stop is a stopping time.

2.

The gambler stops playing when his capital reaches $2,000. The number

τ

=

n

2

of plays that it takes the gambler to stop is a stopping time.

3.

The gambler stops playing when his capital either reaches $2,000, or is depleted, which ever comes first. The number

τ

=

min

⁡

(

n

1

,

n

2

)

of plays that it takes the gambler to stop is a stopping time.

Title

stopping time

Canonical name

StoppingTime

Date of creation

2013-03-22 14:41:13

Last modified on

2013-03-22 14:41:13

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

11

Author

gel (22282)

Entry type

Definition

Classification

msc 60K05

Classification

msc 60G40

Related topic

DoobsOptionalSamplingTheorem

Related topic

PredictableStoppingTime

Generated on Sat Feb 10 12:01:52 2018 by

LaTeXML
Wald’s equation

Wald’s equation

Let

X

1

,

X

2

,

…

,

X

N

be a

sequence

of

N

iid

random variables

distributed as random variable

X

, such that

1.

N

>

0

is itself a random variable (integer-valued),

2.

the

expectation

of

X

,

E

⁡

[

X

]

<

∞

, and

3.

E

⁡

[

N

]

<

∞

.

Then

E

⁡

[

∑

i

=

1

N

X

i

]

=

E

⁡

[

N

]

⁢

E

⁡

[

X

]

.

The integer

N

from above can be viewed as a

stopping time

for the

stochastic process

{

X

i

∣

i

∈

ℤ

+

}

.

Title

Wald’s equation

Canonical name

WaldsEquation

Date of creation

2013-03-22 14:40:04

Last modified on

2013-03-22 14:40:04

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 60K05

Classification

msc 60G40

Generated on Sat Feb 10 12:01:54 2018 by

LaTeXML
bibliography for statistics

bibliography for statistics

References for Statistics, MSC 62 - XX

Introduction, 62A

1.

David Freedman, Robert Pisani, Roger Purves,

Statistics

, 3rd ed., W. W. Norton & Company, 1997.

2.

Morris H. DeGroot, Mark J. Schervish

Probability and Statistics

,
3rd ed., Addison Wesley, 2001.

Foundations, MSC 62A, 62B, 62C

1.

Leonard J. Savage,

The Foundations of Statistics

,
2nd ed., Dover Publications, Inc. New York, 1972.

2.

Robert V. Hogg, Allen T. Craig, Joseph W. McKean,

An Introduction to
Mathematical Statistics

, 6th ed., Prentice Hall, 2004.

3.

George Casella, Roger L. Berger,

Statistical Inference

, 2nd ed.,
Duxbury Press, 2001.

4.

David R. Cox, D. V. Hinkley,

Theoretical Statistics

, Chapman & Hall/CRC,
1979.

5.

Peter J. Bickel, Kjell A. Doksum,

Mathematical Statistics, Volume 1, Basic Ideas
and Selected Topics

, 2rd ed. Prentice Hall, 2001.

6.

T. S. Ferguson,

Mathematical Statistics: A Decision
Theoretic Approach

, Academic Press, Inc., New York, 1967

7.

Harald Cramér,

Mathematical Methods of Statistics

, Princeton, 1946

Sampling Theory, MSC 62D

1.

William G. Cochran,

Sampling Techniques

, 3rd ed., Wiley
Series in Probability and Statistics, John Wiley and Sons, Inc. New
York, 1977.

Linear Inferences, MSC 62J

1.

C. Radhakrishna Rao,

Linear Statistical Inferences and Its Applications

,
2nd ed., Wiley-Interscience, 2002

2.

Douglas C. Montgomery, Elizabeth A. Peck, G. Geoffrey Vining,

Introduction to

Linear Regression

Analysis

, 3rd ed.,
Wiley-Interscience, 2001.

3.

Henry Scheffé,

The

Analysis of Variance

, Wiley-Interscience, 1999.

4.

Peter McCullagh, J. A. Nelder,

Generalized Linear Models

, 2nd ed.,
Monographs on Statistics and Applied Probability 37, Chapman &
Hall/CRC, 1999.

Design of Experiments, MSC 62K

1.

Douglas C. Montgomery,

Design and Analysis of Experiments

,
5th ed., John Wiley and Sons, Inc. New York, 2000.

Multivariate Analysis, MSC 62H

1.

Theodore W. Anderson,

An Introduction to Multivariate
Statistical Analysis

, 3rd ed., Wiley-Interscience, 2003

Title

bibliography for statistics

Canonical name

BibliographyForStatistics

Date of creation

2013-03-22 15:06:31

Last modified on

2013-03-22 15:06:31

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Bibliography

Classification

msc 62-00

Generated on Sat Feb 10 12:02:13 2018 by

LaTeXML
covariance

covariance

The

covariance

of two

random variables

X

1

and

X

2

with mean (

http://planetmath.org/ExpectedValue

)

μ

1

and

μ

2

respectively is defined as

cov

⁢

(

X

1

,

X

2

)

:=

E

⁢

[

(

X

1

-

μ

1

)

⁢

(

X

2

-

μ

2

)

]

.

(1)

The covariance of a random variable

X

with itself is simply the

variance

,

E

⁢

[

(

X

-

μ

)

2

]

.

Covariance captures a

measure

of the

correlation

of two

variables

.

Positive

covariance indicates that as

X

1

increases, so does

X

2

. Negative covariance indicates

X

1

decreases as

X

2

increases and vice versa. Zero covariance can indicate that

X

1

and

X

2

are uncorrelated.

The

correlation coefficient

provides a normalized view of correlation based on covariance:

corr

⁢

(

X

,

Y

)

:=

cov

⁢

(

X

,

Y

)

var

⁢

(

X

)

⁢

var

⁢

(

Y

)

.

(2)

corr

⁢

(

X

,

Y

)

ranges from -1 (for negatively correlated variables) through zero (for uncorrelated variables) to +1 (for positively correlated variables).

While if

X

and

Y

are

independent

we have

corr

⁢

(

X

,

Y

)

=

0

, the latter does not imply the former.

Title

covariance

Canonical

name

Covariance

Date of creation

2013-03-22 12:19:29

Last modified on

2013-03-22 12:19:29

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

9

Author

Koro (127)

Entry type

Definition

Classification

msc 62-00

Synonym

cov

Synonym

correlation

Synonym

correlation coefficient

Related topic

Variance

Generated on Sat Feb 10 12:02:15 2018 by

LaTeXML
geometric random variable

geometric random variable

A

geometric random variable

with

parameter

p

∈

(

0

,

1

]

is one whose

density

distribution function

is given by

f

X

⁢

(

x

)

=

p

⁢

(

1

-

p

)

x

,

x

=

0

,

1

,

2

,

…

This is denoted by

X

∼

G

⁢

e

⁢

o

⁢

(

p

)

.

Notes:

1.

A standard application of geometric random variables is where

X

represents the number of failed Bernoulli trials before the first success.

2.

The

expected value

of a geometric random variable is given by

E

⁢

[

X

]

=

1

-

p

p

, and the

variance

by

V

⁢

a

⁢

r

⁢

[

X

]

=

1

-

p

p

2

3.

The

moment generating function

of a geometric random variable is given by

M

X

⁢

(

t

)

=

p

1

-

(

1

-

p

)

⁢

e

t

.

Title

geometric random variable

Canonical

name

GeometricRandomVariable

Date of creation

2013-03-22 11:54:06

Last modified on

2013-03-22 11:54:06

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

14

Author

mathcam (2727)

Entry type

Definition

Classification

msc 62-00

Classification

msc 60-00

Classification

msc 92-01

Classification

msc 92B05

Synonym

geometric distribution

Generated on Sat Feb 10 12:02:17 2018 by

LaTeXML
random variable

random variable

If

(

Ω

,

𝒜

,

P

)

is a

probability space

, then a

random variable

on

Ω

is a

measurable function

X

:

(

Ω

,

𝒜

)

→

S

to a

measurable space

S

(frequently taken to be the

real numbers

with the standard

measure

). The

law

of a random variable is the

probability measure

P

⁢

X

-

1

:

S

→

ℝ

defined by

P

⁢

X

-

1

⁢

(

s

)

=

P

⁢

(

X

-

1

⁢

(

s

)

)

.

A random variable

X

is said to be

discrete

if the set

{

X

⁢

(

ω

)

:

ω

∈

Ω

}

(i.e. the range of

X

) is finite or countable. A more general version of this definition is as follows: A random variable

X

is discrete if there is a countable subset

B

of the range of

X

such that

P

(

X

∈

B

)

=

1

(Note that, as a countable subset of

ℝ

,

B

is

measurable

).

A random variable

Y

is said to be

if it has a

cumulative distribution function

which is

absolutely continuous

(

http://planetmath.org/AbsolutelyContinuousFunction2

).

Example:

Consider the event of throwing a coin. Thus,

Ω

=

{

H

,

T

}

where

H

is the event in which the coin falls head and

T

the event in which falls tails.
Let

X

=

number of tails in the experiment. Then

X

is a (discrete) random variable.

Title

random variable

Canonical name

RandomVariable

Date of creation

2013-03-22 11:53:10

Last modified on

2013-03-22 11:53:10

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

21

Author

mathcam (2727)

Entry type

Definition

Classification

msc 62-00

Classification

msc 60-00

Classification

msc 11R32

Classification

msc 03-01

Classification

msc 20B25

Related topic

DistributionFunction

Related topic

DensityFunction

Related topic

GeometricDistribution2

Defines

discrete random variable

Defines

continuous random variable

Defines

law of a random variable

Generated on Sat Feb 10 12:02:18 2018 by

LaTeXML
Recursive Z-statistic

Recursive Z-statistic

In respones to: Consider a standard Z-statistic used in

hypothesis testing

. One of the

variables

needed to compute the Z-statistic is the number of

observations

. The problem is that with each additional observation one has to recompute the Z-statistic from scratch. It seems like there is no recursive formulation, e.g. a representation such as
Z(n) = Z(n-1) + new piece of

information

. Is there perhaps an approximate recursive formulation? Any other thoughts?
Thanks.

An example hypothesis test is:

H

0

:

μ

=

μ

0

H

1

:

μ

≠

μ

0

We reject this

hypothesis

if

x

¯

is either greater than or lower than a critical value.
Assuming the critical values do not change all you have to update is

Z

0

.

The

test statistic

is:

Z

0

=

X

¯

-

μ

σ

/

n

Assuming you know

σ

, when you get a new variable

X

n

+

1

you can update

x

¯

using

n

,

X

¯

, and

X

n

+

1

, then recalculate

Z

0

.

Now if you do not know

σ

, and your sample size is large enough to use the

Normal distribution

, you have
to update your

sample variance

,

S

2

. If your sample size is not large enough and you are using the

t-distribution

then
your critical values will change when

n

changes.

To do update

S

without recalculating, you should keep running totals of

∑

i

X

i

and

∑

i

X

i

2

,
so you can update

S

using the computation formula for the sample variance.

Title

Recursive Z-statistic

Canonical

name

RecursiveZstatistic

Date of creation

2013-03-22 19:11:30

Last modified on

2013-03-22 19:11:30

Owner

statsCab (25915)

Last modified by

statsCab (25915)

Numerical id

4

Author

statsCab (25915)

Entry type

Definition

Classification

msc 62-00

Generated on Sat Feb 10 12:02:20 2018 by

LaTeXML
variance

variance

Definition

The

variance

of a real-valued

random variable

X

is

Var

⁡

X

=

𝔼

⁢

[

(

X

-

m

)

2

]

,

m

=

𝔼

⁢

X

,

provided that both

expectations

𝔼

⁢

X

and

𝔼

⁢

[

(

X

-

m

)

2

]

exist.

The variance of

X

is often denoted by

σ

2

⁢

(

X

)

,

σ

X

2

,
or simply

σ

2

.
The

exponent

on

σ

is put there so that the number

σ

=

σ

2

is measured in the same units as the random variable

X

itself.

The quantity

σ

=

Var

⁡

X

is called the

standard deviation

of

X

;
because of the compatibility of the measuring units,
standard deviation is usually the quantity that is quoted
to describe an emprical probability distribution, rather than the variance.

Usage

The variance is a

measure

of the dispersion or variation
of a random variable
about its mean

m

.

It is not always the best measure of dispersion for all random variables,
but compared to other measures,
such as the absolute

mean deviation

,

𝔼

⁢

[

|

X

-

m

|

]

,
the variance is the most tractable analytically.

The variance is closely related to the

𝐋

2

norm for
random variables over a

probability space

.

Properties

1.

The variance of

X

is the second moment of

X

minus
the square of the first moment:

Var

⁡

X

=

𝔼

⁢

[

X

2

]

-

𝔼

⁢

[

X

]

2

.

This

formula

is often used to calculate variance analytically.

2.

Variance is not a

linear function

. It scales quadratically,
and is not affected by shifts in the mean of the

distribution

:

Var

⁡

[

a

⁢

X

+

b

]

=

a

2

⁢

Var

⁡

X

,

for any

a

,

b

∈

ℝ

.

3.

A random variable

X

is constant almost surely if and only
if

Var

⁡

X

=

0

.

4.

The variance can also be characterized as
the minimum of expected squared deviation of a random variable from any point:

Var

⁡

X

=

inf

a

∈

ℝ

⁡

𝔼

⁢

[

(

X

-

a

)

2

]

.

5.

For any two random variables

X

and

Y

whose variances exist,
the variance of the

linear combination

a

⁢

X

+

b

⁢

Y

can be expressed in terms of their

covariance

:

Var

⁡

[

a

⁢

X

+

b

⁢

Y

]

=

a

2

⁢

Var

⁡

X

+

b

2

⁢

Var

⁡

Y

+

2

⁢

a

⁢

b

⁢

Cov

⁡

[

X

,

Y

]

,

where

Cov

⁡

[

X

,

Y

]

=

𝔼

⁢

[

(

X

-

𝔼

⁢

X

)

⁢

(

Y

-

𝔼

⁢

Y

)

]

,
and

a

,

b

∈

ℝ

.

6.

For a random variable

X

, with actual

observations

x

1

,

…

,

x

n

,
its variance is often estimated
empirically with the

sample variance

:

Var

⁡

X

≈

s

2

=

1

n

-

1

⁢

∑

i

=

1

n

(

x

i

-

x

¯

)

2

,

x

¯

=

1

n

⁢

∑

j

=

1

n

x

j

.

Title

variance

Canonical name

Variance

Date of creation

2013-03-22 11:53:46

Last modified on

2013-03-22 11:53:46

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

14

Author

stevecheng (10074)

Entry type

Definition

Classification

msc 62-00

Classification

msc 60-00

Classification

msc 81-00

Classification

msc 83-00

Classification

msc 82-00

Classification

msc 55-00

Related topic

GeometricDistribution2

Related topic

StandardDeviation

Related topic

Covariance

Related topic

MeanSquareDeviation

Generated on Sat Feb 10 12:02:23 2018 by

LaTeXML
probability problem

probability problem

This is in response to the following request:

A parent particle divides into 0,1,or 2 particles with

probabilities

1/4,1/2,1/4.it disappears after splitting.let Xn denotes the number of particles in n-th generations with X0=1.find P(X2¿0) and the probabilities that X1=2 given that X2=1.

http://planetmath.org/?op=getobj;from=requests;id=927

For my first entry I will try to answer the question.

Let

p

0

,

p

1

and

p

2

be the nonzero probabilities of dividing into 0, 1, or 2 particles, and let

X

n

denotes the number of particles at the

n

t

⁢

h

generation.

With

X

0

=

1

, find 1)

P

(

X

2

>

2

)

and 2)

P

(

X

1

=

2

|

X

2

=

1

)

1) After two generations there can be at most

2

2

particles so

P

(

X

2

>

2

)

=

P

(

X

2

=

3

)

+

P

(

X

2

=

4

)

P

(

X

2

=

4

)

=

p

2

2

P

(

X

2

=

3

)

=

2

p

1

p

2

2

Note that if

X

2

=

3

, then

X

2

=

2

.

P

(

X

2

>

2

)

=

p

2

2

(

1

+

2

p

1

)

Using your values I get 3/32.

2) From the definition of

conditional probability

P

(

X

1

|

X

2

)

=

P

⁢

(

X

1

∩

X

2

)

P

⁢

(

X

2

)

First

P

(

X

2

=

1

)

=

p

1

2

+

2

p

0

p

1

p

2

Why?
To get to

X

2

=

1

, at

n

=

1

there are either one or two particles, if there is one particle it remains one at

n

=

2

, and if there were two particles at

n

=

1

, then one has to go to zero and the other one—this can happen two ways.

Finally

P

(

X

1

=

1

∩

X

2

)

=

p

1

p

2

.

P

(

X

1

=

2

|

X

2

=

1

)

=

p

2

p

1

+

2

⁢

p

0

⁢

p

2

Using your values I get 2/3.

Now I have a question for you to think about. What happens in the long run, as

n

→

∞

?

Title

probability problem

Canonical name

ProbabilityProblem

Date of creation

2013-03-22 19:11:21

Last modified on

2013-03-22 19:11:21

Owner

statsCab (25915)

Last modified by

statsCab (25915)

Numerical id

4

Author

statsCab (25915)

Entry type

Definition

Classification

msc 62-01

Generated on Sat Feb 10 12:02:25 2018 by

LaTeXML
statistics on PlanetMath

statistics on PlanetMath

This entry is supposed to become a textbook on

statistics

– or
at least a guide to the statistics-related entries on PlanetMath.
It will refer to related entries (e.g. from

probability

) that
are not listed in the same MSC.

(This is currently a stub article. See also

Textbook projects on PlanetMath

.)

Title

statistics on PlanetMath

Canonical

name

StatisticsOnPlanetMath

Date of creation

2013-03-22 14:48:52

Last modified on

2013-03-22 14:48:52

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Topic

Classification

msc 62-01

Generated on Sat Feb 10 12:02:26 2018 by

LaTeXML