Solution we rst derive c by a geometric argument the

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Unformatted text preview: les) and pdfs (for continuous-type random variables) are usually simpler. This situation is illustrated in Figure 3.5. In this chapter we’ll see that the same scheme holds for joint distributions of multiple random variables. We begin in this section with a brief look at joint CDFs. Let X and Y be random variables on a single probability space (Ω, F , P ). The joint cumulative distribution function (CDF) is the function of two variables defined by FX,Y (uo , vo ) = P {X ≤ uo , Y ≤ vo }. 119 120 CHAPTER 4. JOINTLY DISTRIBUTED RANDOM VARIABLES That is, FX,Y (uo , vo ) is the probability that the random point (X, Y ) falls into the shaded region in the u − v plane, shown in Figure 4.1. v (uo ,v o) u Figure 4.1: Region defining FX,Y (uo , vo ). The joint CDF determines the probabilities of all events concerning X and Y . For example, if R is the rectangular region (a, b] × (c, d] in the plane, then P {(X, Y ) ∈ R} = FX,Y (b, d) − FX,Y (b, c) − FX,Y (a, d) + FX,Y (a, c), (4.1) as illustrated in Figure 4.2. The joint CDF of...
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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 Institute of Technology.

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