A probability distribution of deaths at agexwhen policies are counted instead of lives

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Abstract
It is usual to assume that the probability distribution of deaths at age x is given by or by where qx is the probability of death between exact ages x and (x + 1), E is the number of lives exposed to risk of death in that year of age, and y is the random variable expressing the number of deaths occurring. But it has long been known (see e. g. Hardy, 1882) that these distributions are inapplicable when ‘policies’ are counted in E instead of ‘lives’, and it is thus of some importance to seek the probability distribution of the number of policies falling in by death (called policy-claims, for short, hereafter) at any age x, on the assumption that the corresponding distribution of deaths and the distribution of “duplicates” (i. e. the distribution of unique, duplicate, triplicate, quadruplicate, etc. policies) at age x are known.

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