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Unformatted text preview: butions of X and Y ?” In the discrete case this is easy. The
marginal distributions of X and Y are given by
X
P (X = x) =
P (X = x, Y = y )
y P (Y = y ) = X
x P (X = x, Y = y ) (A.11) 214 APPENDIX A. REVIEW OF PROBABILITY To explain the ﬁrst formula in words, if X = x, then Y will take on some
value y , so to ﬁnd P (X = x) we sum the probabilities of the disjoint events
{X = x, Y = y } over all the values of y .
Formula (A.11) generalizes in a straightforward way to continuous distributions: we replace the sum by an integral and the probability functions by
density functions. If X and Y have joint density fX,Y (x, y ) then the marginal
densities of X and Y are given by
Z
fX (x) = fX,Y (x, y ) dy
Z
fY (y ) = fX,Y (x, y ) dx
(A.12)
The verbal explanation of the ﬁrst formula is similar to that of the discrete
case: if X = x, then Y will take on some value y , so to ﬁnd fX (x) we integrate
the joint density fX,Y (x, y ) over all possible values of y .
Two random variables are said to be independent if for any tw...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).
 Spring '10
 DURRETT
 The Land

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