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By Sheldon Ross

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Sample text

The classical wager of Laplace that the sun will rise tomorrow, for instance, has no place in the theory abeyant a reasonable model of a chance experiment (that one can conceive of being in repeated use); similar objections hold for assigning chances to doomsday predictions of, say, terrorist nuclear attacks on major cities, or the destruction of earth by a meteor, or to assigning chances for the discovery of some hitherto unknown propulsion mechanism, and so on. This is unfortunate and we may be reluctant to give up on instinctive, if unformed, ideas of chance in all kinds of circumstances, whether repeatable or not.

It follows that the constructed set function P is indeed positive, additive, and continuous, at least when restricted to intervals and finite or countable unions of them. We will take it on faith temporarily that these familiar properties of integration are preserved even for integrals over general (Borel) sets on the line (however such integrals are to be interpreted). When the sample space is continuous, the positive, integrable, and normalised function f assumes the rôle played by the discrete probability distribution { pk , k ∈ Z } in the discrete case.

We will explore the subject further in Chapters V and VI. 7 Probabilities in simple cases Our example shows that very rich sample spaces spanning a continuum of outcomes can arise out of experiments with impeccable discrete provenances. In such situations it is simpler to deal directly with the continuous space instead of a clumsy sequence of discrete approximations. C ONTINUOUS SPACES If the outcomes of a conceptual probability experiment take a continuum of values on the real line we shall refer to the space as continuous and identify the sample space Ω with the real line R = (−∞, ∞).

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A First Course in Probability, 5th Ed scanned + Solutions Manual by Sheldon Ross

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