Mathematical Methods for Physics and Engineering - Matematica.NET

30.7 GENERATING FUNCTIONSvariance of both sides of (30.66), and using (30.68), we find( ) 2 ( ) 2 ∂Z∂ZV [ Z(X,Y )] ≈ V [X]+ V [Y ], (30.69)∂X∂Ythe partial derivatives being evaluated at X = µ X and Y = µ Y .30.7 Generating functionsAs we saw in chapter 16, when dealing with particular sets of functions f n ,each member of the set being characterised by a different non-negative integern, it is sometimes possible to summarise the whole set by a single function of adummy variable (say t), called a generating function. The relationship betweenthe generating function and the nth member f n of the set is that if the generatingfunction is expanded as a power series in t then f n is the coefficient of t n .Forexample, in the expansion of the generating function G(z,t) =(1− 2zt + t 2 ) −1/2 ,the coefficient of t n is the nth Legendre polynomial P n (z), i.e.∞∑G(z,t) =(1− 2zt + t 2 ) −1/2 = P n (z)t n .We found that many useful properties of, and relationships between, the membersof a set of functions could be established using the generating function and otherfunctions obtained from it, e.g. its derivatives.Similar ideas can be used in the area of probability theory, and two types ofgenerating function can be usefully defined, one more generally applicable thanthe other. The more restricted of the two, applicable only to discrete integraldistributions, is called a probability generating function; this is discussed in thenext section. The second type, a moment generating function, can be used withboth discrete and continuous distributions and is considered in subsection 30.7.2.From the moment generating function, we may also construct the closely relatedcharacteristic and cumulant generating functions; these are discussed insubsections 30.7.3 and 30.7.4 respectively.n=030.7.1 Probability generating functionsAs already indicated, probability generating functions are restricted in applicabilityto integer distributions, of which the most common (the binomial, the Poissonand the geometric) are considered in this and later subsections. In such distributionsa random variable may take only non-negative integer values. The actualpossible values may be finite or infinite in number, but, for formal purposes,all integers, 0, 1, 2,... are considered possible. If only a finite number of integervalues can occur in any particular case then those that cannot occur are includedbut are assigned zero probability.1157