The conditional density fy x v uo is dened only if uo

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Unformatted text preview: X and Y also determines the probabilities of any v d c + ! ! + u a b Figure 4.2: P {(X, Y ) ∈ shaded region} is equal to FX,Y evaluated at the corners with signs shown. event concerning X alone, or Y alone. To show this, we show that the CDF of X alone is determined by the joint CDF of X and Y. Write +∞ as an argument of FX,Y in place of v to denote the limit as v → +∞. By the countable additivity axiom of probability (Axiom P.2 in Section 1.2), FX,Y (u, +∞) = lim FX,Y (u, v ) = FX (u). v →+∞ Thus, the CDF of X is determined by the joint CDF. Similarly, FY (v ) = FX,Y (+∞, v ). 4.2. JOINT PROBABILITY MASS FUNCTIONS 121 Properties of CDFs The joint CDF, FX,Y , for a pair of random variables X and Y , has the following properties. For brevity, we drop the subscripts on FX,Y and write it simply as F : • 0 ≤ F (u, v ) ≤ 1 for all (u, v ) ∈ R2 • F (u, v ) is nondecreasing in u and is nondecreasing in v • F (u, v ) is right-continuous in u and right-continuous in v • If a < b and c <...
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