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Unformatted text preview: if X and Y are independent, the joint CDF factors: FX,Y (uo , vo ) = FX (uo )FY (vo ). (4.12) Since the CDF completely determines probabilities of the form P {X ∈ A, Y ∈ B }, it turns out that that the converse is also true: If the CDF factors (i.e. (4.12) holds for all uo , vo ). then X and Y are independent (i.e. (4.11) holds for all A, B. In practice, we like to use the generality of (4.11) when doing calculations, but the condition (4.12) is easier to check. To illustrate that (4.12) is stronger than it might appear, suppose that (4.12) holds for all values of uo , vo , and suppose a < b and c < d. By the four point difference formula for CDFs, illustrated in Figure 4.2, P {a < X ≤ b, c < Y ≤ d} = FX (b)FX (d) − FX (b)FX (c) − FX (a)FX (d) + FX (a)FX (c) = (FX (b) − FX (a))(FY (d) − FY (c)) = P {a < X ≤ b}P {c < Y ≤ d}. Therefore, if (4.12) holds for all values of uo , vo , then (4.11) holds whenever A = (a, b] and B = (c, d] for some a, b, c, d. It can be show...
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