Generating functions

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Generating functions
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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.

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• 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.

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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

• 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

• Let us consider a random variable X,
• where the probability, PXjpj
• Suppose X is an integral valued random variable
  with values 0,1,2,.
• Then we can define the tail probabilities as
• PrX gt j
  qj
• The distribution function is thus
• PrXj 1- qj

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

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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
• 2Q(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-r1)
• P(r) (1) rQ (r-1) (1)

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Convolutions

• Consider two non negative independent integral
  valued random variables X and Y, having the
  probability distribution,
• PXj aj and PYk bk
• The probability of the event (Xj,Yk) is
  therefore
• Pr (Xj,Yk) aj bk
•  

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• Suppose we form a new random variable SXY
• Then the event Sr comprises the mutually
  exclusive events,
• (X0,Yr),(X1,Yr-1),.(Xm,Yr-m),
  ..(Xr-1,Y1),(Xr,Y0)
• If the distribution of S is given by PSrcr
• Then it follows that
• Cra0bra1br-1..arb0
• This method of compounding two sequences
• of numbers is called a convolution cjajbj

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

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• ProofSuppose we form a new random variable
  SXY. Let Sn and PSncn
• Then the corresponding generating function is
• Ctd

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• 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 SnX1X2..Xn
• Then the g.f.of Sn is P(x)n

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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
• XY and YX , have the same distribution.
• 3.It is an associative operation. (order is
  immaterial) X(YZ) (XY) Z , have the same
  distribution

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Examples

• Find the p.g.f , the mean and the variance of
• 1.Bernoulli distribution where,
  pPsuccessPX1, and qPX0. Where X
  number of successes in a trial.
• 2.Poisson distribution PXr 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 PXjpqj X?
  define
• 4.Binomial distribution (X_ number of successes
  in n number of trials.) PXr nCr pr q n-r
  ,
• where p Psuccess and q Pfailure

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• 1. For the Bernoulli r.v.
• 
  P(s)qps
• the mean
  P1p
• the variance
  P1P1-P12
• 
  0p-p2p(1-p)
• qp
• 4. For the Binomial r.v. ,which is the sum of n
  independent r.v.s, P(s)qpsn
• Meannp
  
• Variance npq

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• 2. For the Poisson r.v.
• Px
• The mean P1?
• And the varianceP1P1-P12
• ?

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• 3. For the Geometric r.v.
• Px
• The mean Px x1

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• To find the variance,
• The variance P1P1-P12

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Some more examples

• The
• Where the Psuccessp and Pfailureq
• Number of independent trials for the kth success
  Yk, 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

Negative Binomial random variable Y,
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• Let Ykn and yn-k

• The m.g.f is M(? ,t)qpe-?)-k
• And the p.g.f is P(x,t)qpx-1)-k

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• Note that when k1, 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

• Consider the sum of n independent random
  variables, where the number of r.v. contributing
  to the sum is also a r.v.
• Suppose SNX1X2XN
• PrXijfj, PrNngn, PrSNkhk,
• with the corresponding p.g.fs.

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This property is mainly used in discrete
branching processes
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Moment generating function

• Moment generating function (m.g.f) of a r.v. Y
  is defined as M(?)Ee ?Y
• If Y is a discrete integer valued r.v. with
  probability PYjpj
• Then M(?)

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• 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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• 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(?)(qpe?)n
• That is replace x in p.g.f. P(x) by e?..
• M(?)n(qpe?)n-1p
• EX M(?) ?0 n(qpe?)n-1p ?0
• n(qp)n-1p , since e? 1 when ?0
• np
• Similarly it can be shown that V(X)npq