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Unformatted text preview: ∞ The integral in (4.15) can be viewed as the integral of fX,Y over the line u + v = c shown in Figure
4.14. This is an integral form of the law of total probability, because in order for X + Y = c, it
is necessary that there is some value u such that X = u and Y = c − u. The integral in (4.15)
integrates (and integration is a type of summation) over all possible values of u.
If X and Y are independent, so that fX,Y (u, v ) = fX (u)fY (v ), then (4.15) becomes
∞ fX (u)fY (c − u)du. fS (c) = (4.16) −∞ which, by the deﬁnition of the convolution operation “∗”, means
fS = fX ∗ fY (if S = X + Y , where X , Y are independent). Note the strong similarity between (4.13) and (4.14), derived for sums of integervalued random
variables, and (4.15) and (4.16), derived for sums of continuoustype random variables. 4.5. DISTRIBUTION OF SUMS OF RANDOM VARIABLES 139 Example 4.5.4 Suppose X and Y are independent, with each being uniformly distributed over
the interval [0, 1]. Find the pdf of S...
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 Spring '08
 Zahrn
 The Land

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