aarjaneiro/planetmath huggingface dataset:
systematic sampling

systematic sampling

Systematic sampling

is a method of sampling

n

items from a

population

of

N

items with the following

sequence

of procedures:

1.

label each item in the population as

x

i

, where

1

≤

i

≤

N

,

2.

randomly select one item from the first

k

items in the
population, where

k

=

⌈

N

/

n

⌉

, the

smallest integer
greater than or equal to

N

/

n

,

3.

pick the

k

th item thereafter,

x

i

,

x

i

+

k

,

x

i

+

2

⁢

k

,

…

,
until all

n

items are picked.

If

N

≡

0

mod

n

, then all

n

items can be picked before
reaching the end of the population. Otherwise, define

x

j

+

N

:=

x

j

. Using this, we can continue to pick our sample
units until all

n

items are picked.

The above method suggests that we do not have to make our first pick
from among the first

k

items. By the above definition, we can
start anywhere in the population and still end up

n

units of
sample.

Remarks.

•

If

n

divides

N

, then systematic sampling can be viewed as
grouping the population into

k

=

N

/

n

strata, and picking one sample
from each

stratum

. The

difference

between systematic sampling and
stratified sampling is that in systematic sampling, only the first
sample is picked randomly, all other samples are picked based on the
position of the first pick.

•

Again, if

n

∣

N

, one can view systematic sampling as a

one-stage cluster sampling

, where a

primary

sampling unit is defined
as the set of units

x

i

,

x

i

+

k

,

…

,

x

i

+

(

n

-

1

)

⁢

k

in the
population, where

k

=

N

/

n

and

1

≤

i

≤

k

. In this way, there
are

k

primary sampling units. A

simple random sample

of one unit
can then be drawn from these

k

units. All of the items within the
selected primary sample will then be the

complete

sample drawn by
systematic sampling.

•

If some linear trend exists in the population, it is a good
idea to order the population so the linearity is preserved by the
order, so that when a systematic sampling is performed, the
linearity gets carried over to the samples.

•

When ordering the population, care should be taken so that there
does not exhibit any periodic patterns (seasonality, etc…) that
would greatly bias the sample.

Title

systematic sampling

Canonical name

SystematicSampling

Date of creation

2013-03-22 15:15:23

Last modified on

2013-03-22 15:15:23

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

4

Author

CWoo (3771)

Entry type

Definition

Classification

msc 62D05

Generated on Sat Feb 10 12:03:31 2018 by

LaTeXML
exponential random variable

exponential random variable

X

is a

exponential random variable

with

parameter

λ

>

0

if its

probability density function

is given for

x

>

0

by

f

X

⁢

(

x

)

=

λ

⁢

e

-

λ

⁢

x

.

To denote this, one usually writes

X

∼

E

⁢

x

⁢

p

⁢

(

λ

)

.

For an exponential random variable

X

:

1.

X

is commonly used to model lifetimes and duration between Poisson events.

2.

The

expected value

of

X

is given by

E

⁢

[

X

]

=

1

λ

3.

The

variance

of

X

is given by

V

⁢

a

⁢

r

⁢

[

X

]

=

1

λ

2

4.

The moments of

X

are given by

M

X

⁢

(

t

)

=

λ

λ

-

t

5.

It is interesting to note that

X

is a

gamma random variable

with an

α

parameter of 1.

Title

exponential random variable

Canonical

name

ExponentialRandomVariable

Date of creation

2013-03-22 11:54:23

Last modified on

2013-03-22 11:54:23

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Definition

Classification

msc 62E15

Classification

msc 06F20

Classification

msc 11B65

Classification

msc 05C15

Synonym

exponential distribution

Generated on Sat Feb 10 12:03:37 2018 by

LaTeXML
hypergeometric random variable

hypergeometric random variable

X

is a

hypergeometric random variable

with

parameters

M

,

K

,

n

if

f

X

⁢

(

x

)

=

(

K

x

)

⁢

(

M

-

K

n

-

x

)

(

M

n

)

,

x

=

{

0

,

1

,

…

,

n

}

Parameters:

⋆

M

∈

{

1

,

2

,

…

}

⋆

K

∈

{

0

,

1

,

…

,

M

}

⋆

n

∈

{

1

,

2

,

…

,

M

}

Syntax:

X

∼

H

⁢

y

⁢

p

⁢

e

⁢

r

⁢

g

⁢

e

⁢

o

⁢

(

M

,

K

,

n

)

Notes:

1.

X

represents the number of “special” items (from the

K

special items) present on a sample of

n

from a

population

with

M

items.

2.

The

expected value

of

X

is noted as

E

⁢

[

X

]

=

n

⁢

K

M

3.

The

variance

of

X

is noted as

V

⁢

a

⁢

r

⁢

[

X

]

=

n

⁢

K

M

⁢

M

-

K

M

⁢

M

-

n

M

-

1

Approximation techniques:

If

(

K

2

)

≪

n

,

M

-

K

+

1

-

n

then

X

can be approximated as a

binomial

random variable

with parameters

n

=

K

and

p

=

M

-

K

+

1

-

n

M

-

K

+

1

.
This approximation simplifies the

distribution

by looking at a system with replacement for large values of

M

and

K

.

Title

hypergeometric random variable

Canonical

name

HypergeometricRandomVariable

Date of creation

2013-03-22 11:54:12

Last modified on

2013-03-22 11:54:12

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

11

Author

alozano (2414)

Entry type

Definition

Classification

msc 62E15

Classification

msc 81-00

Synonym

hypergeometric distribution

Generated on Sat Feb 10 12:03:39 2018 by

LaTeXML
lognormal random variable

lognormal random variable

X

is a

lognormal random variable

with parameters

μ

∈

ℝ

and

σ

2

>

0

if its

probability density function

is given for

x

>

0

by

f

X

⁢

(

x

)

=

1

2

⁢

π

⁢

σ

2

⁢

e

-

(

ln

⁡

x

-

μ

)

2

2

⁢

σ

2

x

.

To denote this, one usually writes

X

∼

L

⁢

o

⁢

g

⁢

N

⁢

(

μ

,

σ

2

)

.

For a lognormal random variable

X

:

1.

X

is a

random variable

such that

ln

⁡

(

X

)

is a

normal random variable

with mean

μ

and

variance

σ

2

.

2.

E

⁢

[

X

]

=

e

μ

+

σ

2

/

2

3.

V

⁢

a

⁢

r

⁢

[

X

]

=

e

2

⁢

μ

+

σ

2

⁢

(

e

σ

2

-

1

)

4.

M

X

⁢

(

t

)

is not a useful quantity.

Title

lognormal random variable

Canonical name

LognormalRandomVariable

Date of creation

2013-03-22 11:54:46

Last modified on

2013-03-22 11:54:46

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

12

Author

mathcam (2727)

Entry type

Definition

Classification

msc 62E15

Synonym

lognormal distribution

Generated on Sat Feb 10 12:03:41 2018 by

LaTeXML
negative binomial random variable

negative binomial random variable

X

is a

negative binomial random variable

with

parameters

r

and

p

if

f

X

⁢

(

x

)

=

(

r

+

x

-

1

x

)

⁢

p

r

⁢

(

1

-

p

)

x

,

x

=

{

0

,

1

,

…

}

Parameters:

⋆

r

>

0

⋆

p

∈

[

0

,

1

]

Syntax:

X

∼

N

⁢

e

⁢

g

⁢

B

⁢

i

⁢

n

⁢

(

r

,

p

)

Notes:

1.

If

r

∈

ℕ

,

X

represents

the number of failed Bernoulli trials before the

r

th success. Note that if

r

=

1

the

variable

is a

geometric random variable

.

2.

E

⁢

[

X

]

=

r

⁢

1

-

p

p

3.

V

⁢

a

⁢

r

⁢

[

X

]

=

r

⁢

1

-

p

p

2

4.

M

X

⁢

(

t

)

=

(

p

1

-

(

1

-

p

)

⁢

e

t

)

r

Title

negative binomial random variable

Canonical

name

NegativeBinomialRandomVariable

Date of creation

2013-03-22 11:54:15

Last modified on

2013-03-22 11:54:15

Owner

bgins (4516)

Last modified by

bgins (4516)

Numerical id

9

Author

bgins (4516)

Entry type

Definition

Classification

msc 62E15

Classification

msc 18E05

Classification

msc 18-00

Synonym

negative binomial distribution

Generated on Sat Feb 10 12:03:43 2018 by

LaTeXML
negative hypergeometric random variable

negative hypergeometric random variable

X

is a

negative hypergeometric random variable

with

parameters

W

,

B

,

b

if

f

X

⁢

(

x

)

=

(

x

+

b

-

1

x

)

⁢

(

W

+

B

-

b

-

x

W

-

x

)

(

W

+

B

W

)

,

x

=

{

0

,

1

,

…

,

W

}

Parameters:

⋆

W

∈

{

1

,

2

,

…

}

⋆

B

∈

{

1

,

2

,

…

}

⋆

b

∈

{

1

,

2

,

…

,

B

}

Syntax:

X

∼

N

⁢

e

⁢

g

⁢

H

⁢

y

⁢

p

⁢

e

⁢

r

⁢

g

⁢

e

⁢

o

⁢

(

W

,

B

,

b

)

Notes:

1.

X

represents

the number of “special” items (from the

W

special items) present before the

b

th object from a

population

with

B

items.

2.

The

expected value

of

X

is noted as

E

⁢

[

X

]

=

W

⁢

b

B

+

1

3.

The

variance

of

X

is noted as

V

⁢

a

⁢

r

⁢

[

X

]

=

W

⁢

b

⁢

(

B

-

b

+

1

)

⁢

(

W

+

B

+

1

)

(

B

+

2

)

⁢

(

B

+

1

)

2

Approximation techniques:

If

(

x

2

)

≪

W

and

(

b

2

)

≪

B

then

X

can be approximated as a

negative binomial random variable

with parameters

r

=

b

and

p

=

W

W

+

B

.
This approximation simplifies the

distribution

by looking at a system with replacement for large values of

W

and

B

.

Title

negative hypergeometric random variable

Canonical

name

NegativeHypergeometricRandomVariable

Date of creation

2013-03-22 12:25:05

Last modified on

2013-03-22 12:25:05

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

16

Author

alozano (2414)

Entry type

Definition

Classification

msc 62E15

Synonym

negative hypergeometric distribution

Generated on Sat Feb 10 12:03:45 2018 by

LaTeXML
negative hypergeometric random variable, example of

negative hypergeometric random variable, example of

Suppose you have 7 black marbles and 10 white marbles in a jar. You pull marbles until you have 3 black marbles in your hand.

X

would represent the number of white marbles in your hand.

⋆

The

expected value

of

X

would be

E

⁢

[

X

]

=

W

⁢

b

B

+

1

=

3

⁢

(

10

)

7

+

1

=

3.75

⋆

The

variance

of

X

would be

V

⁢

a

⁢

r

⁢

[

X

]

=

W

⁢

b

⁢

(

B

-

b

+

1

)

⁢

(

W

+

B

+

1

)

(

B

+

2

)

⁢

(

B

+

1

)

2

=

10

⁢

(

3

)

⁢

(

7

-

3

+

1

)

⁢

(

10

+

7

+

1

)

(

7

+

2

)

⁢

(

7

+

1

)

2

=

1.875

⋆

The

probability

of having 3 white marbles would be

f

X

⁢

(

3

)

=

(

3

+

b

-

1

3

)

⁢

(

W

+

B

-

b

-

3

W

-

3

)

(

W

+

B

W

)

=

(

3

+

3

-

1

3

)

⁢

(

10

+

7

-

3

-

3

10

-

3

)

(

10

+

7

10

)

=

0.1697

Title

negative hypergeometric random variable

, example of

Canonical name

NegativeHypergeometricRandomVariableExampleOf

Date of creation

2013-03-22 12:39:04

Last modified on

2013-03-22 12:39:04

Owner

aparna (103)

Last modified by

aparna (103)

Numerical id

4

Author

aparna (103)

Entry type

Example

Classification

msc 62E15

Generated on Sat Feb 10 12:03:47 2018 by

LaTeXML
Pareto random variable

Pareto random variable

X

is a

Pareto random variable

with parameters

a

,

b

if

f

X

⁢

(

x

)

=

a

⁢

b

a

x

a

+

1

,

x

∈

[

b

,

∞

)

Parameters:

⋆

a

∈

(

0

,

∞

)

⋆

b

∈

(

0

,

∞

)

Syntax:

X

∼

P

⁢

a

⁢

r

⁢

e

⁢

t

⁢

o

⁢

(

a

,

b

)

Notes:

1.

X

represents a

random variable

with shape parameter

a

and

scale parameter

b

.

2.

The

expected value

of

X

is noted as

E

⁢

[

X

]

=

a

⁢

b

a

-

1

with

a

∈

{

2

,

3

,

…

}

3.

The

variance

of

X

is noted as

V

⁢

a

⁢

r

⁢

[

X

]

=

a

⁢

b

2

(

a

-

1

)

2

⁢

(

a

-

2

)

,

a

∈

{

3

,

4

,

…

}

4.

The

cumulative distribution function

of

X

is noted as

F

⁢

(

x

)

=

1

-

(

b

x

)

a

5.

The moments of

X

around 0 are noted as

E

⁢

[

X

n

]

=

a

⁢

b

n

a

-

n

,

n

∈

{

1

,

2

,

…

,

a

-

1

}

Title

Pareto random variable

Canonical name

ParetoRandomVariable

Date of creation

2013-03-22 12:34:05

Last modified on

2013-03-22 12:34:05

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

9

Author

alozano (2414)

Entry type

Definition

Classification

msc 62E15

Synonym

Pareto distribution

Generated on Sat Feb 10 12:03:51 2018 by

LaTeXML
Poisson random variable

Poisson random variable

The Poisson

discrete probability function

with

parameter

λ

>

0

is given by

f

X

⁢

(

x

)

=

e

-

λ

⁢

λ

x

x

!

,

x

∈

ℕ

.

A

random variable

X

with such a

density

has

expectation

,

variance

,

moment generating function

and

characteristic function

given by

E

⁢

[

X

]

=

λ

,

V

⁢

a

⁢

r

⁢

[

X

]

=

λ

,

M

X

⁢

(

t

)

=

e

λ

⁢

(

e

t

-

1

)

, and

ϕ

X

⁢

(

t

)

=

e

λ

⁢

(

e

i

⁢

t

-

1

)

, respectively.

Title

Poisson random variable

Canonical

name

PoissonRandomVariable

Date of creation

2013-03-22 11:54:03

Last modified on

2013-03-22 11:54:03

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

13

Author

Koro (127)

Entry type

Definition

Classification

msc 62E15

Classification

msc 92B05

Classification

msc 92-01

Synonym

Poisson distribution

Generated on Sat Feb 10 12:03:53 2018 by

LaTeXML
proof of expected value of the hypergeometric distribution

proof of expected value of the hypergeometric distribution

We will first prove a useful property of

binomial coefficients

. We know

(

n

k

)

=

n

!

k

!

⁢

(

n

-

k

)

!

.

This can be transformed to

(

n

k

)

=

n

k

⁢

(

n

-

1

)

!

(

k

-

1

)

!

⁢

(

n

-

1

-

(

k

-

1

)

)

!

=

n

k

⁢

(

n

-

1

k

-

1

)

.

(1)

Now we can start with the definition of the

expected value

:

E

⁢

[

X

]

=

∑

x

=

0

n

x

⁢

(

K

x

)

⁢

(

M

-

K

n

-

x

)

(

M

n

)

.

Since for

x

=

0

we add a

0

in this we can say

E

⁢

[

X

]

=

∑

x

=

1

n

x

⁢

(

K

x

)

⁢

(

M

-

K

n

-

x

)

(

M

n

)

.

Applying equation (

1

) we get:

E

⁢

[

X

]

=

n

⁢

K

M

⁢

∑

x

=

1

n

(

K

-

1

x

-

1

)

⁢

(

M

-

1

-

(

K

-

1

)

n

-

1

-

(

x

-

1

)

)

(

M

-

1

n

-

1

)

.

Setting

l

:=

x

-

1

we get:

E

⁢

[

X

]

=

n

⁢

K

M

⁢

∑

l

=

0

n

-

1

(

K

-

1

l

)

⁢

(

M

-

1

-

(

K

-

1

)

n

-

1

-

l

)

(

M

-

1

n

-

1

)

.

The sum in this equation is

1

as it is the sum over all

probabilities

of a

hypergeometric distribution

. Therefore we have

E

⁢

[

X

]

=

n

⁢

K

M

.

Title

proof of expected value of the hypergeometric distribution

Canonical name

ProofOfExpectedValueOfTheHypergeometricDistribution

Date of creation

2013-03-22 13:27:44

Last modified on

2013-03-22 13:27:44

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

8

Author

mathwizard (128)

Entry type

Proof

Classification

msc 62E15

Generated on Sat Feb 10 12:03:55 2018 by

LaTeXML
proof of variance of the hypergeometric distribution

proof of variance of the hypergeometric distribution

We will first prove a useful property of

binomial coefficients

. We know

(

n

k

)

=

n

!

k

!

⁢

(

n

-

k

)

!

.

This can be transformed to

(

n

k

)

=

n

k

⁢

(

n

-

1

)

!

(

k

-

1

)

!

⁢

(

n

-

1

-

(

k

-

1

)

)

!

=

n

k

⁢

(

n

-

1

k

-

1

)

.

(1)

The

variance

Var

⁡

[

X

]

of

X

is given by:

Var

⁡

[

X

]

=

∑

x

=

0

n

(

x

-

n

⁢

K

M

)

2

⁢

(

K

x

)

⁢

(

M

-

K

n

-

x

)

(

M

n

)

.

We expand the right hand side:

Var

⁡

[

X

]

=

∑

x

=

0

n

x

2

⁢

(

K

x

)

⁢

(

M

-

K

n

-

x

)

(

M

n

)

-

2

⁢

n

⁢

K

M

⁢

∑

x

=

0

n

x

⁢

(

K

x

)

⁢

(

M

-

K

n

-

x

)

(

M

n

)

+

n

2

⁢

K

2

M

2

⁢

∑

x

=

0

n

(

K

x

)

⁢

(

M

-

K

n

-

x

)

(

M

n

)

.

The second of these sums is the

expected value

of the

hypergeometric distribution

, the third sum is

1

as it sums up all

probabilities

in the

distribution

. So we have:

Var

⁡

[

X

]

=

-

n

2

⁢

K

2

M

2

+

∑

x

=

0

n

x

2

⁢

(

K

x

)

⁢

(

M

-

K

n

-

x

)

(

M

n

)

.

In the last sum for

x

=

0

we add nothing so we can write:

Var

⁡

[

X

]

=

-

n

2

⁢

K

2

M

2

+

∑

x

=

1

n

x

2

⁢

(

K

x

)

⁢

(

M

-

K

n

-

x

)

(

M

n

)

.

Applying equation (

1

) and

x

=

(

x

-

1

)

+

1

we get:

Var

⁡

[

X

]

=

-

n

2

⁢

K

2

M

2

+

n

⁢

K

M

⁢

∑

x

=

1

n

(

x

-

1

)

⁢

(

K

-

1

x

-

1

)

⁢

(

M

-

K

n

-

x

)

(

M

-

1

n

-

1

)

+

n

⁢

K

M

⁢

∑

x

=

1

n

(

K

-

1

x

-

1

)

⁢

(

M

-

K

n

-

x

)

(

M

-

1

n

-

1

)

.

Setting

l

:=

x

-

1

the first sum is the expected value of a hypergeometric distribution and is therefore given as

(

n

-

1

)

⁢

(

K

-

1

)

M

-

1

. The second sum is the sum over all the probabilities of a hypergeometric distribution and is therefore equal to

1

. So we get:

Var

⁡

[

X

]

=

-

n

2

⁢

K

2

M

2

+

n

⁢

K

⁢

(

n

-

1

)

⁢

(

K

-

1

)

M

⁢

(

M

-

1

)

+

n

⁢

K

M

=

-

n

2

⁢

K

2

⁢

(

M

-

1

)

+

M

⁢

n

⁢

(

n

-

1

)

⁢

K

⁢

(

K

-

1

)

+

K

⁢

n

⁢

M

⁢

(

M

-

1

)

M

2

⁢

(

M

-

1

)

=

n

⁢

K

⁢

(

M

2

+

(

-

K

-

n

)

⁢

M

+

n

⁢

K

)

M

2

⁢

(

M

-

1

)

=

n

⁢

K

⁢

(

M

-

K

)

⁢

(

M

-

n

)

M

2

⁢

(

M

-

1

)

=

n

⁢

K

M

⁢

(

1

-

K

M

)

⁢

M

-

n

M

-

1

.

This is the one we wanted to prove.

Title

proof of variance of the hypergeometric distribution

Canonical name

ProofOfVarianceOfTheHypergeometricDistribution

Date of creation

2013-03-22 13:27:41

Last modified on

2013-03-22 13:27:41

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

13

Author

mathwizard (128)

Entry type

Proof

Classification

msc 62E15

Generated on Sat Feb 10 12:03:59 2018 by

LaTeXML
proof that normal distribution is a distribution

proof that normal distribution is a distribution

∫

-

∞

∞

e

-

(

x

-

μ

)

2

2

⁢

σ

2

σ

⁢

2

⁢

π

⁢

𝑑

x

=

(

∫

-

∞

∞

e

-

(

x

-

μ

)

2

2

⁢

σ

2

σ

⁢

2

⁢

π

⁢

𝑑

x

)

2

=

∫

-

∞

∞

e

-

(

x

-

μ

)

2

2

⁢

σ

2

σ

⁢

2

⁢

π

⁢

𝑑

x

⁢

∫

-

∞

∞

e

-

(

y

-

μ

)

2

2

⁢

σ

2

σ

⁢

2

⁢

π

⁢

𝑑

y

=

∫

-

∞

∞

∫

-

∞

∞

e

-

(

x

-

μ

)

2

+

(

y

-

μ

)

2

2

⁢

σ

2

σ

2

⁢

2

⁢

π

⁢

𝑑

x

⁢

𝑑

y

Substitute

x

′

=

x

-

μ

and

y

′

=

y

-

μ

. Since the bounds are infinite, they do not change, and

d

⁢

x

′

=

d

⁢

x

and

d

⁢

y

′

=

d

⁢

y

. Thus, we have

∫

-

∞

∞

∫

-

∞

∞

e

-

(

x

-

μ

)

2

+

(

y

-

μ

)

2

2

⁢

σ

2

σ

2

⁢

2

⁢

π

⁢

𝑑

x

⁢

𝑑

y

=

∫

-

∞

∞

∫

-

∞

∞

e

-

(

x

′

)

2

+

(

y

′

)

2

2

⁢

σ

2

σ

2

⁢

2

⁢

π

⁢

𝑑

x

′

⁢

𝑑

y

′

.

Converting to polar coordinates, we obtain

∫

-

∞

∞

∫

-

∞

∞

e

-

(

x

′

)

2

+

(

y

′

)

2

2

⁢

σ

2

σ

2

⁢

2

⁢

π

⁢

𝑑

x

′

⁢

𝑑

y

′

=

∫

0

∞

∫

0

2

⁢

π

r

⁢

e

-

r

2

2

⁢

σ

2

σ

2

⁢

2

⁢

π

⁢

𝑑

r

⁢

𝑑

θ

=

∫

0

2

⁢

π

d

⁢

θ

2

⁢

π

⁢

∫

0

∞

r

⁢

e

-

r

2

2

⁢

σ

2

σ

2

⁢

𝑑

r

=

θ

2

⁢

π

|

0

2

⁢

π

⁢

1

σ

2

⁢

∫

0

∞

r

⁢

e

-

r

2

2

⁢

σ

2

⁢

𝑑

r

=

2

⁢

π

2

⁢

π

⁢

σ

2

σ

2

⁢

(

-

e

-

r

2

2

⁢

σ

2

)

|

0

∞

=

1

⁢

1

=

1

.

Title

proof that

normal distribution

is a

distribution

Canonical name

ProofThatNormalDistributionIsADistribution

Date of creation

2013-03-22 13:29:12

Last modified on

2013-03-22 13:29:12

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

7

Author

Wkbj79 (1863)

Entry type

Proof

Classification

msc 62E15

Related topic

AreaUnderGaussianCurve

Generated on Sat Feb 10 12:04:03 2018 by

LaTeXML
properties of Poisson random variables

properties of Poisson random variables

Proposition 1

.

If

X

1

,

X

2

are

independent

Poisson random variables

with parameters

λ

1

,

λ

2

, then

X

1

+

X

2

is a Poisson random variable with parameter

λ

1

+

λ

2

.

Proof.

Let

X

:=

X

1

+

X

2

and

λ

:=

λ

1

+

λ

2

, let us calculate the

distribution function

of

X

:

F

X

⁢

(

x

)

=

P

(

X

≤

x

)

=

P

(

X

1

+

X

2

≤

x

)

=

∑

i

=

0

x

P

(

X

1

+

X

2

=

i

)

=

∑

i

=

0

x

∑

j

=

0

i

P

(

X

1

=

j

and

X

2

=

i

-

j

)

=

∑

i

=

0

x

∑

j

=

0

i

P

(

X

1

=

j

)

P

(

X

2

=

i

-

j

)

=

∑

i

=

0

x

∑

j

=

0

i

e

-

λ

1

⁢

λ

1

j

j

!

⁢

e

-

λ

2

⁢

λ

2

i

-

j

(

i

-

j

)

!

=

∑

i

=

0

x

∑

j

=

0

i

e

-

λ

i

!

⁢

(

i

j

)

⁢

λ

1

j

⁢

λ

2

i

-

j

=

∑

i

=

0

x

e

-

λ

i

!

⁢

∑

j

=

0

i

(

i

j

)

⁢

λ

1

j

⁢

λ

2

i

-

j

=

∑

i

=

0

x

e

-

λ

i

!

⁢

(

λ

1

+

λ

2

)

i

=

∑

i

=

0

x

e

-

λ

i

!

⁢

λ

i

.

As a result,

X

is a Poisson random variable with parameter

λ

. Notice that in the fifth equation, we used the assumption that

X

1

and

X

2

are independent.
∎

As a corollary, any sum of independent Poisson random variables is Poisson, with parameter the sum of the parameters from the independent

random variables

.

Proposition 2

.

A Poisson random variable is

infinitely divisible

.

Proof.

Let

X

be a Poisson random variable with parameter

λ

. Let

n

be any

positive integer

. Let

X

1

,

…

,

X

n

be

independent identically distributed

Poisson random variables with parameter

λ

n

. Then the sum of these random variables is easily seen to be Poisson, with parameter

λ

, and is therefore identically distributed as

X

.
∎

Title

properties of Poisson random variables

Canonical name

PropertiesOfPoissonRandomVariables

Date of creation

2013-03-22 18:50:55

Last modified on

2013-03-22 18:50:55

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

4

Author

CWoo (3771)

Entry type

Derivation

Classification

msc 62E15

Generated on Sat Feb 10 12:04:13 2018 by

LaTeXML
unit normal loss function

unit normal loss function

The

function,

U

⁢

N

⁢

L

, is defined by

U

⁢

N

⁢

L

⁢

(

c

)

=

∫

c

∞

(

t

-

c

)

⁢

f

⁢

(

t

)

⁢

𝑑

t

where

c

is a constant and

f

(

.

)

is the normal

probability distribution function

.

An alternative computational formula for

U

⁢

N

⁢

L

is the following:

U

⁢

N

⁢

L

⁢

(

z

)

=

f

⁢

(

z

)

-

z

⁢

(

1

-

F

⁢

(

z

)

)

where

f

(

.

)

and

F

(

.

)

are the probability distribution function and

cumulative distribution function

for

Standard Normal Distribution

respectively.

Remark.

This function has an extensive use in Risk Analysis and the Theory of Blackjack.

Title

unit normal loss function

Canonical name

UnitNormalLossFunction

Date of creation

2013-03-22 15:56:42

Last modified on

2013-03-22 15:56:42

Owner

georgiosl (7242)

Last modified by

georgiosl (7242)

Numerical id

6

Author

georgiosl (7242)

Entry type

Definition

Classification

msc 62E15

Generated on Sat Feb 10 12:04:28 2018 by

LaTeXML
Lehmann-Scheffé theorem

Lehmann-Scheffé theorem

A

statistic

S

⁢

(

𝑿

)

on a random sample of data

𝑿

=

(

X

1

,

…

,

X

n

)

is said to be a

complete statistic

if for any

Borel measurable function

g

,

E

(

g

(

S

)

)

=

0

implies

P

(

g

(

S

)

=

0

)

=

1

.

In other words,

g

⁢

(

S

)

=

0

almost everywhere whenever the

expected value

of

g

⁢

(

S

)

is

0

. If

S

⁢

(

𝑿

)

is associated with a family

f

(

x

∣

θ

)

of

probability density functions

(or

mass function

in the discrete case), then completeness of

S

means that

g

⁢

(

S

)

=

0

almost everywhere whenever

E

θ

⁢

(

g

⁢

(

S

)

)

=

0

for every

θ

.

Theorem 1

(Lehmann-Scheffé).

If

S

⁢

(

𝐗

)

is a

complete

sufficient statistic

and

h

⁢

(

𝐗

)

is an

unbiased estimator

for

θ

, then, given

h

0

(

s

)

=

E

(

h

(

𝑿

)

|

S

(

𝑿

)

=

s

)

,

h

0

⁢

(

S

)

=

h

0

⁢

(

S

⁢

(

𝑿

)

)

is a

uniformly minimum variance unbiased

estimator

of

θ

. Furthermore,

h

0

⁢

(

S

)

is unique almost everywhere for every

θ

.

Title

Lehmann-Scheffé theorem

Canonical name

LehmannScheffeTheorem

Date of creation

2013-03-22 16:31:59

Last modified on

2013-03-22 16:31:59

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 62F10

Synonym

Lehmann-Scheffe theorem

Defines

complete statistic

Generated on Sat Feb 10 12:04:42 2018 by

LaTeXML