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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 continuoustype random variables Suppose S = X + Y where X and Y are jointly continuoustype. 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.
 Spring '08
 Zahrn
 The Land

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