51 suppose x has the binomial distribution with

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Unformatted text preview: ts A, B, and C are mutually independent if they are pairwise independent, and P (ABC ) = P (A)P (B )P (C ). The general definition of mutual independence for n random variables was given in Section 2.4.2. Namely, X1 , X2 , . . . , Xn are mutually independent if any set of events of the form {X1 ∈ A1 }, {X2 ∈ A2 }, . . . , {Xn ∈ An } are mutually independent. In this section we cover in more detail the special case of independence for two 132 CHAPTER 4. JOINTLY DISTRIBUTED RANDOM VARIABLES random variables, although factorization results are given which can be extended to the case of n mutually independent random variables. 4.4.1 Definition of independence for two random variables Definition 4.4.1 Random variables X and Y are defined to be independent if any pair of events of the form {X ∈ A} and {Y ∈ B }, are independent. That is: P {X ∈ A, Y ∈ B } = P {X ∈ A}P {Y ∈ B }. (4.11) Taking A and B to be sets of the form A = {u : u ≤ uo } and B = {v : v ≤ vo } shows that...
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