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).
Sums of random number of random variables (random sums). Let X1;X2;X3;:::: be a sequence of independent identically distributed random variables (i.i.d. random variables), each with the same distribution, each having common mean a = E(X) and variance s2 =Var(X). Here X is a r.v. having the same distribution as Xj. The sum S =åN j=1 Xj
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