Events a b and c are mutually independent if they are

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Unformatted text preview: ut to X instead of to u, so E [Y |X ] is a random variable. Conditional expectations play an important role in estimation of one random variable from another, as shown in Section 4.10. 4.3. JOINT PROBABILITY DENSITY FUNCTIONS 125 Example 4.3.1 Suppose X and Y have the joint pdf: c(1 − u − v ) if u ≥ 0, v ≥ 0, u + v ≤ 1 0 else, fX,Y (u, v ) = (4.8) where c is a constant to be determined. The pdf and its support are shown in Figure 4.4. Find the v f (u,v) 1 X,Y v support u u 1 1 Figure 4.4: The pdf (4.8) and its support. constant c, the marginal pdf of X, and the conditional pdf of Y given X. Solution: We first derive c by a geometric argument. The value of c is such that the integral of fX,Y over the plane is one. Equivalently, the volume of the region between the u − v plane and the graph of fX,Y should be one. That region is similar to a cone or pyramid, in that it is the space between a two-dimensional base and a vertex. The volume of such regions is one third the area of the base times the height of the vertex above the plane of...
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