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Introduction to Probability phần 8

Giá thích hợp cho một cuộc sống tiền trợ cấp hàng năm là giá trị dự kiến của tiền trợ cấp hàng năm thiết bị đầu cuối được đánh giá cho các đời ngẫu nhiên của người mua. Vì vậy, công việc của Huygens trong việc giới thiệu giá trị kỳ vọng và làm việc của Graunt và Halley trong việc xác định các bảng tỷ lệ tử vong | 9.2. DISCRETE INDEPENDENT TRIALS 349 a population is that the variance does not seem to increase or decrease from one generation to the next. This was known at the time of Galton and his attempts to explain this led him to the idea of regression to the mean. This idea will be discussed further in the historical remarks at the end of the section. The reason that we only consider one sex is that human heights are clearly sex-linked and in general if we have two populations that are each normally distributed then their union need not be normally distributed. Using the multiple-gene hypothesis it is easy to explain why the variance should be constant from generation to generation. We begin by assuming that for a specific gene location there are k alleles which we will denote by A1 A2 . Ak. We assume that the offspring are produced by random mating. By this we mean that given any offspring it is equally likely that it came from any pair of parents in the preceding generation. There is another way to look at random mating that makes the calculations easier. We consider the set S of all of the alleles at the given gene location in all of the germ cells of all of the individuals in the parent generation. In terms of the set S by random mating we mean that each pair of alleles in S is equally likely to reside in any particular offspring. The reader might object to this way of thinking about random mating as it allows two alleles from the same parent to end up in an offspring but if the number of individuals in the parent population is large then whether or not we allow this event does not affect the probabilities very much. For 1 i k we let Pi denote the proportion of alleles in the parent population that are of type Aj. It is clear that this is the same as the proportion of alleles in the germ cells of the parent population assuming that each parent produces roughly the same number of germs cells. Consider the distribution of alleles in the offspring. Since each germ cell