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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 integer-valued random
variables, and (4.15) and (4.16), derived for sums of continuous-type 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
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