Actuarial Mathematics and Life-Table Statistics
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This book is a course of lectures on the mathematics of actuarial science. The idea behind the lectures is as far as possible to deduce interesting material on contingent present values and life tables directly from calculus and common- sense notions, illustrated through word problems. Both the Interest Theory and Probability related to life tables are treated as wonderful concrete appli- cations of the calculus. The lectures require no background beyond a third semester of calculus, but the prerequisite calculus courses must have been solidly understood. It is a truism of pre-actuarial advising that students who have not done really well in and digested the calculus ought not to consider actuarial studies.
It is not assumed that the student has seen a formal introduction to prob- ability. Notions of relative frequency and average are introduced ¯rst with reference to the ensemble of a cohort life-table, the underlying formal random experiment being random selection from the cohort life-table population (or, in the context of probabilities and expectations for `lives aged x', from the subset of lx members of the population who survive to age x). The cal- culation of expectations of functions of a time-to-death random variables is rooted on the one hand in the concrete notion of life-table average, which is then approximated by suitable idealized failure densities and integrals. Later, in discussing Binomial random variables and the Law of Large Numbers, the combinatorial and probabilistic interpretation of binomial coe±cients are de- rived from the Binomial Theorem, which the student the is assumed to know as a topic in calculus (Taylor series identi¯cation of coe±cients of a poly- nomial.) The general notions of expectation and probability are introduced, but for example the Law of Large Numbers for binomial variables is treated (rigorously) as a topic involving calculus inequalities and summation of ¯nite series. This approach allows introduction of the numerically and conceptually useful large-deviation inequalities for binomial random variables to explain just how unlikely it is for binomial (e.g., life-table) counts to deviate much percentage-wise from expectations when the underlying population of trials is large. Download free Actuarial Mathematics and Life-Table Statistics.pdf here
It is not assumed that the student has seen a formal introduction to prob- ability. Notions of relative frequency and average are introduced ¯rst with reference to the ensemble of a cohort life-table, the underlying formal random experiment being random selection from the cohort life-table population (or, in the context of probabilities and expectations for `lives aged x', from the subset of lx members of the population who survive to age x). The cal- culation of expectations of functions of a time-to-death random variables is rooted on the one hand in the concrete notion of life-table average, which is then approximated by suitable idealized failure densities and integrals. Later, in discussing Binomial random variables and the Law of Large Numbers, the combinatorial and probabilistic interpretation of binomial coe±cients are de- rived from the Binomial Theorem, which the student the is assumed to know as a topic in calculus (Taylor series identi¯cation of coe±cients of a poly- nomial.) The general notions of expectation and probability are introduced, but for example the Law of Large Numbers for binomial variables is treated (rigorously) as a topic involving calculus inequalities and summation of ¯nite series. This approach allows introduction of the numerically and conceptually useful large-deviation inequalities for binomial random variables to explain just how unlikely it is for binomial (e.g., life-table) counts to deviate much percentage-wise from expectations when the underlying population of trials is large. Download free Actuarial Mathematics and Life-Table Statistics.pdf here
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