By V. S. Pugachev

ISBN-10: 0080291481

ISBN-13: 9780080291482

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**Extra info for Probability Theory and Mathematical Statistics for Engineers**

**Sample text**

It is interesting to calculate the probability of the union of non-exclusive events. 12) for two events A\= A and A2 = B we get P(AUB) = P(A)+P(BÄ). 9) B = BAUBÄ, and therefore P(B) = P(AB)+P(BÄ). Solving this for P(BÄ) and substituting it into the preceding equality we obtain P(AUB) = P(A)+P(B)-P(AB). 7. Complete set of events. A countable set of events {Ak} is called a complete set of events if at least one of them appears as a result of a trial. In other words, the events Al9 . . , An9 n =*s <~, form a complete set if their union is a certain event : U Ak = Ω.

22) /=i Thus the probability of the event A is equal to the sum of the probabilities of the events Hl9 . . , Hn multiplied by the respective conditional probabilities of the event A. 22) is called a. formula of total probability. It is widely used in probability theory and its applications. 17. e. for every nx devices produced by thefirstplant there are nz devices produced by the second plant and n3 devices produced by the third plant). Suppose that the trademark on the devices is absent and that the rate of failure (constant) for the devices from the first plant is λΐ9 for the devices from the second plant is A2 and for the devices from the third plant is λ3.

Thus, the prob- 24 Probability Theory and Mathematical Statistics for Engineers ability of any event belongs to the interval [0,1]: O^P(A)^ 1. After representing the union of events Al9 . . , An as the union of exclusive events : n U A i = A1ÖA2Ä1{JA3Ä:iÄ2U . . UAnÄ! . 11) we obtain / > ( | J Λ/) = Ρ(Αί)+Ρ(Α2Α1)~{-Ρ(Α3Α~1Α^ + . . Α^Ϊ). 12) But Α2Αι c: A2, A3A1A2 a As, . . , AnA\ . . An-\

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