Mathematical Methods for Physics and Engineering - Matematica.NET

30.7 GENERATING FUNCTIONSComparing this expression with (30.92), we find that κ 1 = µ, κ 2 = σ 2 and all othercumulants are equal to zero. ◭We may obtain expressions for the cumulants of a distribution in terms of itsmoments by differentiating (30.92) with respect to t to givedK X= 1 dM X.dt M X dtExpanding each term as power series in t and cross-multiplying, we obtaint(κ 2···)(1 + κ 2 t + κ 32! + t 2···)1+µ 1 t + µ 22! + t=(µ 2···)1 + µ 2 t + µ 32! + ,and, on equating coefficients of like powers of t on each side, we findµ 1 = κ 1 ,µ 2 = κ 2 + κ 1 µ 1 ,µ 3 = κ 3 +2κ 2 µ 1 + κ 1 µ 2 ,µ 4 = κ 4 +3κ 3 µ 1 +3κ 2 µ 2 + κ 1 µ 3 ,.µ k = κ k + k−1 C 1 κ k−1 µ 1 + ···+ k−1 C r κ k−r µ r + ···+ κ 1 µ k−1 .Solving these equations for the κ k , we obtain (for the first four cumulants)κ 1 = µ 1 ,κ 2 = µ 2 − µ 2 1 = ν 2 ,κ 3 = µ 3 − 3µ 2 µ 1 +2µ 3 1 = ν 3 ,κ 4 = µ 4 − 4µ 3 µ 1 +12µ 2 µ 2 1 − 3µ2 2 − 6µ4 1 = ν 4 − 3ν2 2 . (30.93)Higher-order cumulants may be calculated in the same way but become increasinglylengthy to write out in full.The principal property of cumulants is their additivity, which may be provedby combining (30.92) with (30.90). If X 1 , X 2 , ..., X N are independent randomvariables and K Xi (t) for i =1, 2,...,N is the CGF for X i then the CGF ofS N = c 1 X 1 + c 2 X 2 + ···+ c N X N (where the c i are constants) is given byK SN (t) =N∑K Xi (c i t).i=1Cumulants also have the useful property that, under a change of origin X →X + a the first cumulant undergoes the change κ 1 → κ 1 + a but all higher-ordercumulants remain unchanged. Under a change of scale X → bX, cumulant κ rundergoes the change κ r → b r κ r .1167