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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
 

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[X|A] = 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[X||X| = x] =
fX(x) - fX(-x)
fX(x) + fX(-x)
|x|, if x > 0 and fX(x) + fX(-x) > 0,
E[X||X| = 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(A|G2), for each A G3.
(ii) P(B C|G2) = P(B|G2)P(C|G2), 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)}

  

Source: Arcones, Miguel A. - Department of Mathematical Sciences, State University of New York at Binghamton

 

Collections: Mathematics