Transcription of Probability 2 - Notes 5 Conditional expectations E X Y as ...
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Probability 2 - Notes 5 Conditional expectations E X Y as ...
Probability 2 - Notes 5 Conditional expectationsE(X|Y)as random variablesConditional expectations were discussed in lectures (see also the second part of Notes 3). Thegoal of these Notes is to provide a summary of what has been done so far. We start by remindingthe main definitions and by listing several results which were proved in lectures (and Notes 3).LetXandYbe two discrete s with a joint ,Y(x,y) =P(X=x,Y=y). Rememberthat the distributions (or the s)fX(x) =P(X=x)ofXandfY(y) =P(Y=y)ofYarecalled the marginal distributions of the pare(X,Y)and thatfX(x) = yfX,Y(x,y)andfY(y) = xfX,Y(x,y).IffY(y)6=0, the Conditional ofX|Y=yis given byfX|Y(x|y)def=fX,Y(x,y)fY(y)and the condi-tional expectation byE(X|Y=y)def= xx fX|Y(x|y)and, more generally,E(g(X)|Y=y)def= xg(x)fX|Y(x|y),is defined for any real valued functiong(X). In particular,E(X2|Y=y)is obtained wheng(X) =X2andVar(X|Y=y) =E(X2|Y=y) [E(X|Y=y)] always suppose that x|g(x)|fX|Y(x|y) . (y) =E(X|Y=y). ThenE(X|Y)def= (Y).
Probability 2 - Notes 5 Conditional expectations E(XjY) as random variables Conditional expectations were discussed in lectures (see also the second part of Notes 3). The
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