By D.R. Cox
This ebook will be of curiosity to senior undergraduate and postgraduate scholars of utilized facts.
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Extra info for Applied Statistics: Principles and Examples (Chapman & Hall Statistics Text Series)
Suppose also that it is reasonable to believe that the outcome of either experiment should not change the probability of any event in the other experiment. We can then produce a probability model for the combination of experiments as follows. Let S = S1 × S2 , and for (s1 , s2 ) ∈ S, let P((s1 , s2 )) = P1 (s1 )P2 (s2 ). In this way we have deﬁned a probability measure on S. To see this, note that for any events, A1 ⊂ S1 , A2 ⊂ S2 , P(A1 × A2 ) = P((s1 , s2 )) = s1 ∈A1 ,s2 ∈A2 P1 (s1 ) s1 ∈A1 P2 (s2 ) s2 ∈A2 = P1 (A1 )P2 (A2 ).
Let p1 = P(A1 ) and p2 = P(A2 ) for one experiment. Prove that P(A1 occurs before A2 ) = p1 /( p1 + p2 ). Hint: Let the sample space S consist of all ﬁnite sequences of the form (B, B, . . , B, Ai ) for i = 1 or 2. (b) Use a conditional probability argument to ﬁnd the probability that the player wins a game. RANDOM VARIABLES Suppose that two dice are thrown. Let M denote the maximum of the numbers appearing on the two dice. For example, for the outcome (4, 5), M is 5, and for the outcome (3, 3) M is 3.
X n. That is, independence of X 1 , . . , X n is equivalent to f (k1 , . . , kn ) = f 1 (k1 ) f 2 (k2 ) · · · f n (kn ) for every (k1 , . . , kn ) ∈ Rn . In the same way, if (Si , Pi ) is a probability model for experiment E i for i = 1, . . , n, (S, P) is their product model, and for each i, X i depends only on the outcome of E i (only on the outcome si ∈ Si ), then X 1 , . . , X n are independent. 7 Box k has k balls, numbered 1, . . , k, for k = 2, 3, 4. Balls are chosen independently from these boxes, one per box.