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Advanced Level Mathematics: Statistics 2 by Steve Dobbs, Jane Miller, Julian Gilbey

By Steve Dobbs, Jane Miller, Julian Gilbey

Written to check the contents of the Cambridge syllabus. facts 2 corresponds to unit S2. It covers the Poisson distribution, linear combos of random variables, non-stop random variables, sampling and estimation, and speculation exams.

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Example text

Examination of the works of Bernoulli, DeMoivre, and Laplace does not entirely bear out this claim. There the equality of chances appears as a mathematical device, not a metaphysical necessity. Nevertheless, the contrast of chance with “art,” “design,” or “cause,” that “something, which gives it superiority,” is a recurrent theme. De Moivre suggests that “we may imagine Chance and Design to be, as it were, in Competition with each other” (De Moivre 1756, p. v). “Chance” and “Design” here no longer mean the presence and absence of a stochastic element, but a lack of uniformity in the probability distribution.

In both formal logic and subjective probability, the theory polices sets of beliefs by testing them for inconsistencies, but it does not dictate how detected inconsistencies should be removed. If, as was the case with some of the symmetry assumptions previously discussed, the consequences are deemed unacceptable, then the assumption will be discarded. If, on the other hand, the axioms seem compelling, as in mathematical probability, then surprising consequences such as the birthday paradox will be regarded as valuable correctives to our erroneous, untutored intuitions; that is why the theory is useful.

If this assumption be admitted, and if it can be shown that there is only one intelligible law, then that must be the actual law. . A person who should dispute the propriety of dividing our belief equally amongst hypotheses about which we are equally ignorant, ought to be refuted by asking him to state which is to be preferred. He must either admit the proposed law, or maintain that there is no law at all. This observation would not have disarmed Ellis, Boole, or Venn, who indeed denied the existence of any (determinate in the case of Boole) law at all.

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