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Generating functions

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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  1. Generating functions MA 4030-probability generating fnctions

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

  7. The probability generating function is • The generating function, • is not a p.g.f. since . MA 4030-probability generating fnctions

  8. 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

  9. 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

  10. 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

  11. 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

  12. Define the associated probability generating functions of the sequences defined earlier. MA 4030-probability generating fnctions

  13. 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

  14. MA 4030-probability generating fnctions

  15. 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 MA 4030-probability generating fnctions

  16. 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

  17. 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

  18. 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 MA 4030-probability generating fnctions

  19. 2. For the Poisson r.v. • P[x]= • The mean =P’[1]= • And the variance=P’’[1]+P’[1]-{P’{1]}2 • = MA 4030-probability generating fnctions

  20. 3. For the Geometric r.v. • P[x]= • The mean =P’[x] | x=1= MA 4030-probability generating fnctions

  21. To find the variance, • The variance =P’’[1]+P’[1]-{P’{1]}2 MA 4030-probability generating fnctions

  22. 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

  23. 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 MA 4030-probability generating fnctions

  24. 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

  25. 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

  26. This property is mainly used in discrete branching processes MA 4030-probability generating fnctions

  27. 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

  28. 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

  29. 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

  30. 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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