# Generating functions will be used at several points

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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).

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