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### Generating functions

MA 4030-probability generating fnctions

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

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

- 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

- 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:

- 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

The probability generating function is

- The generating function,
- is not a p.g.f. since .

MA 4030-probability generating fnctions

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)

MA 4030-probability generating fnctions

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

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]

MA 4030-probability generating fnctions

Generating 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

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

MA 4030-probability generating fnctions

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…

MA 4030-probability generating fnctions

- 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

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

- 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

- 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

- P[x]=
- The mean =P’[1]=
- And the variance=P’’[1]+P’[1]-{P’{1]}2
- =

MA 4030-probability generating fnctions

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

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

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.

MA 4030-probability generating fnctions

This property is mainly used in discrete branching processes

MA 4030-probability generating fnctions

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()=

MA 4030-probability generating fnctions

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))

MA 4030-probability generating fnctions

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.

MA 4030-probability generating fnctions

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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