Probability and Stochastic Processes Lecture Notes, Lecture notes of Probability and Stochastic Processes

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Discrete Probability Law If the sample space consists of a finite number of possible outcomes, then the probability law is specified by the probabilities of the events that consist of a single element. In particular, the probability of any event {s1, 82,...,$n} is the sum of the probabilities of its elements: P({si, 52,...,8n}) = P(s1) + P(s2) +--: + P(sn)- Discrete Uniform Probability Law If the sample space consists of n possible outcomes which are equally likely (i-e., all single-element events have the same probability), then the proba- bility of any event A is given by P(A) = number of elements of A n Some Properties of Probability Laws Consider a probability law, and let A, B, and C be events. (a) If AC B, then P(A) < P(B). (b) P(AU B) = P(A) + P(B) — P(AN B). (c) P(AU B) < P(A) + P(B). (d) P(AUBUC) = P(A) + P(ACN B) + P(ACN BENC).