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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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Tech.
 Spring '08
 Zahrn
 The Land

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