By Michael J. Evans - Probability and Statistics: The by M.J. Evans

By M.J. Evans

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2 depicts a decreasing sequence of subsets. 2: A decreasing sequence of subsets A1 m A2 m A3 m     We will consider such sequences of sets at several points in the text. For this we need the following result. 1 Let A A1  A2     be events, and suppose that either An  A or An  A. 7 for the proof of this theorem. 1 Suppose S is the set of all positive integers, with Ps  2s for all s + S. Then what is P 5 6 7 8    ? , An  A. 6: Continuity of P have P 5 6 7 8       lim P 5 6 7 8     n  n* lim P5  P6      Pn n* t 5 u s 2  2n1 25  26      2n  lim n* n* 1  21 r s  lim 24  2n  24  116 r  lim n* Alternatively, we could use countable additivity directly, to conclude that P 5 6 7 8      P5  P6  P7      which amounts to the same thing.

1 (Law of total probability, discrete random variable version) Let X be a discrete random variable, and let A be some event. Then ; PX  x PA X  x. 1 Important Discrete Distributions Certain particular discrete distributions are so important that we list them here. 1 Degenerate Distributions Let c be some fixed real number. Then, as already discussed, c is also a random variable (in fact, c is a constant random variable). In this case, clearly c is discrete, with probability function pc satisfying that pc c  1, and pc x  0 for x / c.

On the other hand, we similarly compute that Pfirst coin tails two coins heads  13 We thus see that conditioning on some event (such as “two coins heads”) can make probabilities either increase (as for the event “first coin heads”) or decrease (as for the event “first coin tails”). 1, the law of total probability, in a different and sometimes more helpful way. 1 (Law of total probability, conditioned version) Let A1  A2     be events that form a partition of the sample space S, each of positive probability.

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