Outline

Probability theory 2008

Outline

The need for transforms

Probability-generating function

Moment-generating function

Characteristic function

Applications of transforms to branching processes


Definition of transform

In probability theory, a transform is function that uniquely determines the probability distribution of a random variable

An example:

..

.

.

)0(')1(

)0()0(

10,)()()(0

gXP

gXP

tnXPttEtgn

nXX


Using transforms to determine the distribution of a sum of random variables

YX TT and

YXT

YX ff and

YXf


The probability generating function

Let X be an integer-valued nonnegative random variable. The probability generating function of X is

Defined at least for | t | < 1 Determines the probability function of X uniquely Adding independent variables corresponds to multiplying their generating functions

Example 1: X Be(p)

Example 2: X Bin(n;p)

Example 3: X Po(λ)

Addition theorems for binomial and Poisson distributions

0

)()()(n

nXX nXPttEtg


The moment generating function

Let X be a random variable. The moment generating function of X is

provided that this expectation is finite for | t | < h, where h > 0

Determines the probability function of X uniquely Adding independent variables corresponds to multiplying their moment

generating functions

)()( tXX eEt


The moment generating functionand the Laplace transform

Let X be a non-negative random variable. Then

)()()()()(0

)(

0

tLdxxfedxxfeeEt XXxt

XtxtX

X


The moment generating function- examples

The moment generating function of X is

Example 1: X Be(p)

Example 2: X Exp(a)

Example 3: X (2;a)

)()( tXX eEt


The moment generating function- calculation of moments

)(!

...)()()(0

k

k

k

XtxtX

X XEk

tdxxfeeEt

)0(!

)( )(

0

kX

k

k

X k

tt


The moment generating function- uniqueness

...,2,1,0,)()()()( kdxxfxdxxfxtt Yk

Xk

YX

)()()( where...,2,1,0,0)( xfxfxhkdxxhx YXk


Normal approximation of a binomial distribution

Let X1, X2, …. be independent and Be(p) and let

Then

But

n

npXXY n

n

...1

n

nntntp

nntntpY

non

ppt

epe

pepetn

)/1(2

)1(1

))1(1(

)1()(

2

//

/

nasen

a an)1(


The characteristic function

Let X be a random variable. The characteristic function of X is

Exists for all random variables Determines the probability function of X uniquely Adding independent variables corresponds to multiplying their

characteristic functions

)(sin)(cos)()( tXiEtXEeEt itXX


Comparison of the characteristic function and the moment generating function

Example 1: Exp(λ)

Example 2: Po(λ)

Example 3: N( ; )

Is it always true that

.

)()( itt XX


The characteristic function- uniqueness

For discrete distributions we have

For continuous distributions with

we have

.

dttX |)(|

)()(2

1xfdtte XX

itx

)()(2

1xXPdtte

T X

T

T

itx


The characteristic function- calculation of moments

If the k:th moment exists we have

.

)()0()( kkkX XEi


Using a normal distribution to approximate a Poisson distribution

Let XPo(m) and set

Then

.

Xm

mm

mXY

1

...)( tY


Using a Poisson distribution to approximate a Binomial distribution

Let XBin(n ; p)

Then

If p = 1/n we get.

nitX pept )1()(

))exp(1()( itetX


Sums of a stochastic number of stochastic variables

Probability generating function:

Moment generating function:

Characteristic function:

NN XXS ...1


Branching processes

Suppose that each individual produces j new offspring with probability pj, j ≥ 0, independently of the number produced by any other individual.

Let Xn denote the size of the nth generation

Then

where Zi represents the number of offspring of the ith individual of the (n - 1)st generation.

1

1

nX

iin ZX

generation


Generating function of a branching processes

Let Xn denote the number of individuals in the n:th generation of a population, and assume that

where Yk, k = 1, 2, … are i.i.d. and independent of Xn

Then

Example:

nX

kkn YX

X

11

0 1

))((...)(1

tggtg YXX nn

tp

pppttg

k

kkY )1(1

)1()(0


Branching processes- mean and variance of generation size

Consider a branching process for which X0 = 1, and and respectively depict the expectation and standard deviation of the offspring distribution.

Then

.

nn

nnnn

XE

ZEXEXXEEXE

)(

)()(...)]|([)( 11

1if,

1if,1

1)(

)]|([)]|([)(

2

12

11

n

XVar

XXEVarXXVarEXVarn

n

n

nnnnn


Branching processes- extinction probability

Let 0 = P(population dies out) and assume that X0 = 1

Then

where g is the probability generating function of the offspring distribution

jj

jj

j

ppjXP

0

010

0 )|out dies population(

)1('

)( 00

g

g


Exercises: Chapter III

3.1, 3.2, 3.3, 3.7, 3.15, 3.25, 3.26, 3.27, 3.32

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