Schaum's Outline of Calculus, 5th ed. (Schaum's Outline Series)
Outline
22
Probability theory 2008 Outline The need for transforms Probability-generating function Moment-generating function Characteristic function Applications of transforms to branching processes
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Outline. The need for transforms Probability-generating function Moment-generating function Characteristic function Applications of transforms to branching processes. Definition of transform. - PowerPoint PPT Presentation
Transcript of Outline
Probability theory 2008
Outline
The need for transforms
Probability-generating function
Moment-generating function
Characteristic function
Applications of transforms to branching processes
Probability theory 2008
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
Probability theory 2008
Using transforms to determine the distribution of a sum of random variables
YX TT and
YXT
YX ff and
YXf
Probability theory 2008
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
Probability theory 2008
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
Probability theory 2008
The moment generating functionand the Laplace transform
Let X be a non-negative random variable. Then
)()()()()(0
)(
0
tLdxxfedxxfeeEt XXxt
XtxtX
X
Probability theory 2008
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
Probability theory 2008
The moment generating function- calculation of moments
)(!
...)()()(0
k
k
k
XtxtX
X XEk
tdxxfeeEt
)0(!
)( )(
0
kX
k
k
X k
tt
Probability theory 2008
The moment generating function- uniqueness
...,2,1,0,)()()()( kdxxfxdxxfxtt Yk
Xk
YX
)()()( where...,2,1,0,0)( xfxfxhkdxxhx YXk
Probability theory 2008
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(
Probability theory 2008
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
Probability theory 2008
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
Probability theory 2008
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
Probability theory 2008
The characteristic function- calculation of moments
If the k:th moment exists we have
.
)()0()( kkkX XEi
Probability theory 2008
Using a normal distribution to approximate a Poisson distribution
Let XPo(m) and set
Then
.
Xm
mm
mXY
1
...)( tY
Probability theory 2008
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
Probability theory 2008
Sums of a stochastic number of stochastic variables
Probability generating function:
Moment generating function:
Characteristic function:
NN XXS ...1
Probability theory 2008
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
Probability theory 2008
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
Probability theory 2008
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
Probability theory 2008
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
Probability theory 2008
Exercises: Chapter III
3.1, 3.2, 3.3, 3.7, 3.15, 3.25, 3.26, 3.27, 3.32
