 
Summary: Math571. 7th Homework. Due Tuesday, May 15, 2007.
1. Find a probability space (, F, P), a r.v. X on and a subfield A of F such that
E[XA] = 0 a.s., but it is not true that X = 0 a.s.
2. Suppose that X1, X2 and X3 are independent r.v.'s. Show that E[X1X3(X1, X2)] =
X1E[X3].
3. Let X be an absolutely continuous r.v. with pdf fX. Show that
E[XX = x] =
fX(x)  fX(x)
fX(x) + fX(x)
x, if x > 0 and fX(x) + fX(x) > 0,
E[XX = 0] = 0
4. Let (, F, P) be a probability space. Let Gi, for i = 1, 2, 3 be three fields of F. Show
that the following conditions are equivalent:
(i) P(A(G1, G2)) = P(AG2), for each A G3.
(ii) P(B CG2) = P(BG2)P(CG2), for each B G1 and each C G3.
5. Let {(Xn, Fn)}
n=1 be a martingale. Prove that {(Xn, Gn)}
n=1 is a martingale, where
Gn = (X1, . . . , Xn).
6. {(Xn, Fn)}
