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Unformatted text preview: n that (4.11) also holds when A and B are finite unions of intervals, and then for all choices of A and B. Recall that for discrete-type random variables it is usually easier to work with pmfs, and for jointly continuous-type random variables it is usually easier to work with pdfs, than with CDFs. Fortunately, in those instances, independence is also equivalent to a factorization property for a joint pmf or pdf. Therefore, discrete-type random variables X and Y are independent if and only if the joint pmf factors: pX,Y (u, v ) = pX (u)pY (v ), for all u, v. And for jointly continuous-type random variables, X and Y are independent if and only if the joint pdf factors: fX,Y (u, v ) = fX (u)fY (v ). 4.4.2 Determining from a pdf whether independence holds Suppose X and Y are jointly continuous with joint pdf fX,Y . So X and Y are independent if and only if fX,Y (u.v ) = fX (u)fY (v ) for all u and v. It takes a little practice to be able to tell, given a choice of fX,Y , whether independence holds. Some propositions are given in this section th...
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