19 simplies to fr c 2cfxy c 0 if fxy is circularly

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Unformatted text preview: er the set {1, 2, 3, 4, 5, 6} and they are independent. The convolution formula for the pmf of S = X + Y is an instance of the fact that the convolution of two identical rectangle functions results in a triangle shaped function. See Fig. 4.13. pX 1/6 pY 1/6 1/6 = * 12 3456 pS 12 3456 2 3 4 5 6 7 8 9 10 11 12 Figure 4.13: The pmf for the sum of the numbers showing for rolls of two fair dice. 138 4.5.2 CHAPTER 4. JOINTLY DISTRIBUTED RANDOM VARIABLES Sums of jointly continuous-type random variables Suppose S = X + Y where X and Y are jointly continuous-type. We will express the pdf fS in terms of the joint pdf, fX,Y . The method will be to first find the CDF of Y and then differentiate it to get the pdf. For any c ∈ R, the event {S ≤ c} is the same as the event that the random point (X, Y ) in the plane falls into the shaded region of Figure 4.14. The shaded region can be integrated v u+v=c u Figure 4.14: Shaded region for computation of FS (c), where S = X + Y. over by integrating over all u, and for each u fixed, integrate over v from −∞ to c − u, so ∞ c−u FS (c) = P {S ≤ c} = fX,Y (u, v )dv du. −∞ −∞ Therefore, fS (c) = dFS (c) = dc ∞ −∞ d dc c−u fX,Y (u, v )dv du −∞ ∞ fX,Y (u, c − u)du. = (4.15)...
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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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