Generating functions
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Generating functions. Generating functions. Definition: Let a 0 ,a 1 ,……a n ,….be a sequence of real numbers and let If the series converges in some real interval (-x 0 , x 0 ), |x|≤ x 0 the function A(x) is called a generating function for {a j }.

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Generating functions

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Generating functions

Generating functions

MA 4030-probability generating fnctions


Generating functions1

Generating functions

  • Definition: Let a0,a1,……an,….be a sequence of real numbers and let

  • If the series converges in some real interval (-x0, x0), |x|≤ x0 the function A(x) is called a generating function for {aj}.

MA 4030-probability generating fnctions


Generating functions

  • The generating function may be regarded as a transformation which carries the sequence {aj} into A(x).In general x will be a real number. However, it is possible to work with the complex numbers as well.

  • If, the sequence {aj}. is bounded , then a comparison with the geometric series shows that A(x) converges at least for |x| ≤1.

MA 4030-probability generating fnctions


Probability generating function

Probability Generating function

  • If we have the additional property that:

    aj ≥0 and

  • then A(x) is called a probability generating function.

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Proposition

Proposition

  • A generating function uniquely determines its sequence.

  • This single function A(x) can be used to represent the whole collection of individual items {aj}.

  • Uses of probability generating functions

    To find the density/mass function

    To find the Moments in stochastic models

    To Calculate limit distributions

    In difference equations or recursions

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Example

Example:

  • Let us consider a random variable X,

    where the probability, P[X=j]=pj

  • Suppose X is an integral valued random variable with values 0,1,2,…….

  • Then we can define the tail probabilities as

    Pr[X > j] = qj

  • The distribution function is thus

    Pr[X≤j] = 1- qj

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Generating functions

  • The probability generating function is

  • The generating function,

  • is not a p.g.f. since .

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Some useful results

Some useful results:

  • 1) 1-P(x) = (1-x) (Q(x))

  • 2.) P’(1) = Q(1)

  • 3.) P’’(1) = 2Q’(1)

  • 4.) V(x)=P’’(1)+P’(1)-[P’(1)]2

    =2’Q’(1) +Q(1)-[Q(1)]2

  • 5) the rth factorial moment or rth moment about the origin,

  • µ’(r) = E [X(X-1) (X-2)……(X-r+1)

  • = P(r) (1) = rQ (r-1) (1)

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Convolutions

Convolutions

  • Consider two non negative independent integral valued random variables X and Y, having the probability distribution,

  • P[X=j] = aj and P[Y=k] = bk

  • The probability of the event (X=j,Y=k) is therefore

  • Pr[ (X=j,Y=k)]= aj bk

MA 4030-probability generating fnctions


Generating functions

  • Suppose we form a new random variable S=X+Y

  • Then the event S=r comprises the mutually exclusive events,

    (X=0,Y=r),(X=1,Y=r-1),….(X=m,Y=r-m), ………..(X=r-1,Y=1),(X=r,Y=0)

    If the distribution of S is given by P[S=r]=cr

    Then it follows that :

    Cr=a0br+a1br-1+…………..+arb0

    This method of compoundingtwo sequences

    of numbers is called a convolution {cj}={aj}{bj]

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Generating functions and convolutions

Generating functions and Convolutions:

  • Proposition : The generating function of a convolution ({cj} ) is the product of the generating functions ({aj},{bj}) .

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Generating functions

  • Define the associated probability generating functions of the sequences defined earlier.

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Generating functions

  • Proof:Suppose we form a new random variable S=X+Y. Let S=n and P[S=n]=cn

  • Then the corresponding generating function is

  • Ctd…

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Generating functions

MA 4030-probability generating fnctions


Generating functions

  • Extensions to convolution:

  • More generally the generating function of {aj},{bj},{cj},…is the product of the generating functions of {aj},{bj},{cj},….

  • F(x)= A(x).B(x).C(x)….

  • Also let X1,X2,…..,Xn be i.i.d r.v.s

  • and Sn=X1+X2+…..+Xn

  • Then the g.f.of Sn is [P(x)]n

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Some properties of convolution

Some properties of convolution

1.The convolution of two probability mass functions on the non negative integers is a pr. Mass function.

2.Convolution is a commutative operation

X+Y and Y+X , have the same distribution.

3.It is an associative operation. (order is immaterial) X+(Y+Z) = (X+Y)+ Z , have the same distribution

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Examples

Examples

  • Find the p.g.f , the mean and the variance of :

    1.Bernoulli distribution where, p=P[success]=P[X=1], and q=P[X=0]. Where X= number of successes in a trial.

    2.Poisson distribution P[X=r]= e-µ µr/r!

    eg: X -. number of phone calls in a unit time interval. µ is the average number of phone calls/unit

    3.Geometric distribution P[X=j]=pqj X? define

    4.Binomial distribution (X_ number of successes in n number of trials.) P[X=r] = nCr pr q n-r ,

    where p= P[success] and q= P[failure]

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Generating functions

  • 1. For the Bernoulli r.v.

  • P(s)=[q+ps]

    the mean =P’[1]=p

    the variance =P’’[1]+P’[1]-{P’{1]}2

    =0+p-p2=p(1-p)

    =qp

    4. For the Binomial r.v. ,which is the sum of n independent r.v.s, P(s)=[q+ps]n

    Mean=np

    Variance =npq

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Generating functions

2. For the Poisson r.v.

  • P[x]=

  • The mean =P’[1]=

  • And the variance=P’’[1]+P’[1]-{P’{1]}2

  • =

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Generating functions

  • 3. For the Geometric r.v.

  • P[x]=

  • The mean =P’[x] | x=1=

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Generating functions

  • To find the variance,

  • The variance =P’’[1]+P’[1]-{P’{1]}2

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Some more examples

Some more examples:

Negative Binomial random variable Y,

  • The :

    Where the P{success]=p and P[failure]=q

  • Number of independent trials for the kth success : Y+k, Or the number of failures before the kth success r.v. Y;

  • This distribution is called the negative binomial because he probabilities correspond to the successive terms of the binomial expansion of

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Generating functions

  • Let Y+k=n and y=n-k

  • The m.g.f is M( ,t)={q+pe-)}-k

  • And the p.g.f is P(x,t)=={q+px-1)}-k

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Generating functions

  • Note that when k=1, the probability mass function becomes qyp, which is the Geometric distribution.

  • It can be shown that: P(x)=

  • Which is the nth power of the p.g.f. of the geometric distribution.

  • Then it is clear that negative binomial r.v. is the convolution of n geometric random variables.

  • Its mean

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Compound distributions

Compound Distributions

  • Consider the sum of n independent random variables, where the number of r.v. contributing to the sum is also a r.v.

  • Suppose SN=X1+X2+…+XN

  • Pr[Xi=j]=fj, Pr[N=n]=gn, Pr[SN=k]=hk,

  • with the corresponding p.g.fs.

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Generating functions

This property is mainly used in discrete branching processes

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Moment generating function

Moment generating function

  • Moment generating function (m.g.f) of a r.v. Y is defined as M()=E[e Y]

  • If Y is a discrete integer valued r.v. with probability P[Y=j]=pj

  • Then M()=

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Generating functions

  • Taylor expansion of of M() generates the

  • moments given by

  • M() =

  • is the rth moment about the origin,

  • M’()|  = =. =E(X), M’’ ()|  =0 =(E(X(X-1))

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Generating functions

  • If X is a continuous r.v.with a frequency distribution f(u), then we have

  • And all the other properties as in the discrete case.

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M g f of the binomial distribution

M.g.f of the Binomial distribution

  • M()=(q+peθ)n

  • That is replace x in p.g.f. P(x) by eθ..

  • M’()=n(q+peθ)n-1p

  • E[X}=M’()| =0 =n(q+peθ)n-1p| =0

  • = n(q+p)n-1p , since eθ=1 when θ=0

  • =np

  • Similarly it can be shown that V(X)=npq

MA 4030-probability generating fnctions


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