(a) E(X1+X2|Y)=E(X1|Y)+E(X2|Y)

(b) E(1|Y)=1

(c) E(X|X)=X

(d) ifX1 X1 thenE(X1|Y)E(X2|Y)

(e) E(X|1)=E(X)

(f) E(E(X|Y))=E(X)

(g) if:RRisaBorelfunctionthen E((Y)X | Y) = (Y)E(X | Y), provided E(|(Y)X|) <

(h) if X and Y are independent then E(X | Y) = E(X)

Using the properties (a)-(h) above show that

(a) IfA,B,A,B,andAandBare disjoint, then

P(A | Y) + P(B | Y) = P(A B | Y).

(b) Forallwehave

0 P(A | Y)() 1.

(c) Forallwehave P(A | Y)() + P(Ac | Y)() = 1, where Ac =\A,isthecomplementof A.