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0ATA adjusted death rate age distribution age groups age interval age of childbearing age-specific death rates approximation Assume assumption average be~rxl(x birth rate calculate census Chapter characteristic equation constant corresponding matrix crude death rate denoted difference differential equation digraph dl(x equal error expj expression female fertility force of mortality formula fraction functional iteration geometric geometric progression given graph Hence hyperbola indirect method initial integral intrinsic rate irreducible iteration Keyfitz logarithm logistic curve LPRINT Makeham's male Markov chain Mathematics matrix of Problem mean age Metropolitan Statistical Area Muncie number of births number of children number of persons obtain percent population aged primitive Problem 17 Problem 22 projection matrix Prove quadratic random variable rate of increase REA0 rural Society of Actuaries solution stable population standard population subpopulations survivorship t)dt ultimate ratio urban values variance vector zero