John E. Freund's Mathematical Statistics with Applications by Irwin Miller

By Irwin Miller

John E. Freund's Mathematical facts with Applications , 8th Edition, presents a calculus-based creation to the idea and alertness of facts, in keeping with finished insurance that displays the newest in statistical considering, the educating of information, and present practices.

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Extra info for John E. Freund's Mathematical Statistics with Applications (8th Edition)

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With reference to the preceding example, suppose that we are interested in the following question: If a rental car delivered to the consulting firm needs an oil change, what is the probability that it came from rental agency 2? To answer questions of this kind, we need the following theorem, called Bayes’ theorem: 46 Probability THEOREM 13. If B1 , B2 , . . , Bk constitute a partition of the sample space S and P(Bi ) Z 0 for i = 1, 2, . . , k, then for any event A in S such that P(A) Z 0 P(Br |A) = P(Br ) · P(A|Br ) k P(Bi ) · P(A|Bi ) i=1 for r = 1, 2, .

Sample spaces and events, particularly relationships among events, are often depicted by means of Venn diagrams, in which the sample space is represented by a rectangle, while events are represented by regions within the rectangle, usually by circles or parts of circles. For instance, the shaded regions of the four Venn diagrams of Figure 3 represent, respectively, event A, the complement of event A, the union of events A and B, and the intersection of events A and B. When we are dealing with three events, we usually draw the circles as in Figure 4.

Solution Since A = {HHH, HHT} B = {HHT, HTT, THT, TTT} C = {HTT, THT, TTH} A ∩ B = {HHT} B ∩ C = {HTT, THT} 42 Probability the assumption that the eight possible outcomes are all equiprobable yields P(A) = 14 , P(B) = 12 , P(C) = 38 , P(A ∩ B) = 18 , and P(B ∩ C) = 14 . (a) Since P(A) · P(B) = 1 4 (b) Since P(B) · P(C) = pendent. · 12 = 1 2 · 3 8 1 8 = P(A ∩ B), events A and B are independent. = 3 16 Z P(B ∩ C), events B and C are not inde- In connection with Definition 5, it can be shown that if A and B are independent, then so are A and B , A and B, and A and B .

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