GERRODETTE ET AL.: CONFIDENCE LIMITS FOR POPULATION PROJECTIONS 



APPENDIX 2. 



A recursive algorithm for computing the higher order 

 moments of the normal distribution. 



The moment generating function for the normal distribution is 



2 



where ^ is the mean and v is the variance of the normal variate x. The nth moment of x is found by 

 evaluating, at ^ = 0, the nth derivative ofM^{t) with respect to t. The wth differentiation with respect to t 

 leads to the series 



(m + vO"M^(0 + ... + A(m+ vt)" v'^ M^{t) + 5(m + vty-^v^^^M^it) + ..., 



which, evaluated oXt = 0, gives 



where A and B are coefficients and a and ft are exponents such that a + 2/3 = w. The next [(n + l)th] dif- 

 ferentiation of the middle terms gives 



A(m + vtf^^ v^M,{t) + Aa{yL + vty-^ v^^^ MJt) 

 + B{yL + vt)"-^ v^*^ M,{t) + B{q - 2)(m + vt)'-^ v^^^ M^(t) 

 = ... + (Aa + 5)(m + vty-^ v'^^^ M,{t) + ... 

 which, evaluated at ^ = 0, gives 



... + (Aa + 5V"^ v^^l + ... . 



Thus the coefficient of each term of the series of the {n + l)th moment can be computed from the two 

 terms in the series of the nth moment "before" and "after" it. The exponents of ^ and v follow the regular 

 pattern shown. 



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