- #1
kingwinner
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" The law of total expectation is: E(Y) = E[E(Y|X)].
It can be generalized to the vector case: E(Y) = E[E(Y|X1,X2)].
Further extension:
(i) E(Y|X1) = E[E(Y|X1,X2)|X1]
(ii) E(Y|X1,X2) = E[E(Y|X1,X2,X3)|X1,X2] "
====================
I understand the law of total expectation itself, but I don't understand the generalizations to the vector case and the extensions.
1) Is E(Y|X1,X2) a random variable? Is E(Y|X1,X2) a function of both X1 and X2? i.e. E(Y|X1,X2)=g(X1,X2) for some function g?
2) Are (i) and (ii) direct consequences of the law of total expectation? (are they related at all?) I don't see how (i) and (ii) can be derived as special cases from it...can somone please show me how?
Any help is much appreciated!
It can be generalized to the vector case: E(Y) = E[E(Y|X1,X2)].
Further extension:
(i) E(Y|X1) = E[E(Y|X1,X2)|X1]
(ii) E(Y|X1,X2) = E[E(Y|X1,X2,X3)|X1,X2] "
====================
I understand the law of total expectation itself, but I don't understand the generalizations to the vector case and the extensions.
1) Is E(Y|X1,X2) a random variable? Is E(Y|X1,X2) a function of both X1 and X2? i.e. E(Y|X1,X2)=g(X1,X2) for some function g?
2) Are (i) and (ii) direct consequences of the law of total expectation? (are they related at all?) I don't see how (i) and (ii) can be derived as special cases from it...can somone please show me how?
Any help is much appreciated!