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Unformatted text preview: EE 505 B, Autumn, 2011 Two or More Random Variables 1 1 Two Random Variables 1.1 Finding the joint cdf from the joint pdf Example 1.1. Find the cdf of the pdf we used in class. The pdf is f X 1 X 2 ( x 1 , x 2 ) = braceleftBigg 2 x 2 x 1 1 all other values of x 1 and x 2 . The region of the nonzero portion of the pdf is shown in Figure 1 . x 1 x 2 1 1 Figure 1: The region in which the pdf is nonzero. The cdf is found by integrating the pdf. F X 1 X 2 ( x 1 , x 2 ) = integraldisplay x 1- integraldisplay x 2- f X 1 X 2 ( 1 , 2 ) d 2 d 1 The region of integration is the part of the two-dimension space that is to the left of x 1 and below x 2 . The integral will only result in a nonzero value when the region of integration overlaps the nonzero part of the pdf. In general, the form of the function that describes the cdf will change when the geometry (shape) of the intersection of the region of integration with the nonzero portion of the pdf changes. Figure 2 shows the intersection of the region of integration with the nonzero portion of the pdf for all the...
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