# Applied Statistics: Principles and Examples (Chapman & Hall by D.R. Cox

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)**

**Example text**

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.