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# Generating functions - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about ' Generating functions' - marcell-connor

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### Generating functions

MA 4030-probability generating fnctions

• 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

• 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 which carries the sequence {a

• If we have the additional property that:

aj ≥0 and

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

MA 4030-probability generating fnctions

Proposition which carries the sequence {a

• 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

MA 4030-probability generating fnctions

Example: which carries the sequence {a

• 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

MA 4030-probability generating fnctions

MA 4030-probability generating fnctions

Some useful results: which carries the sequence {a

• 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)

MA 4030-probability generating fnctions

Convolutions which carries the sequence {a

• 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

• Suppose we form a new random variable S=X+Y which carries the sequence {a

• 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]

MA 4030-probability generating fnctions

Generating which carries the sequence {a functions and Convolutions:

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

MA 4030-probability generating fnctions

MA 4030-probability generating fnctions

• Proof: the sequences defined earlier. 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…

MA 4030-probability generating fnctions

MA 4030-probability generating fnctions the sequences defined earlier.

• Extensions to convolution: the sequences defined earlier.

• 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

MA 4030-probability generating fnctions

Some properties of convolution the sequences defined earlier.

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

MA 4030-probability generating fnctions

Examples the sequences defined earlier.

• 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]

MA 4030-probability generating fnctions

• 1. For the Bernoulli r.v. the sequences defined earlier.

• 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

MA 4030-probability generating fnctions

2. For the Poisson r.v. the sequences defined earlier.

• P[x]=

• The mean =P’[1]=

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

• =

MA 4030-probability generating fnctions

MA 4030-probability generating fnctions

MA 4030-probability generating fnctions

Some more examples: the sequences defined earlier.

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

MA 4030-probability generating fnctions

• The m.g.f is M( ,t)={q+pe-)}-k

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

MA 4030-probability generating fnctions

• 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

MA 4030-probability generating fnctions

• 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.

MA 4030-probability generating fnctions

MA 4030-probability generating fnctions

Moment generating function processes

• 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()=

MA 4030-probability generating fnctions

• Taylor expansion of of M( processes) generates the

• moments given by

• M() =

• is the rth moment about the origin,

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

MA 4030-probability generating fnctions

MA 4030-probability generating fnctions

M.g.f of the Binomial distribution then we have

• 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