Download STAT 6371 – Probability Theory

STAT 6371 – Probability Theory
Assignment #8
Due November 22, 2005
1. If X n  0 for all n and X n  X with probability one, then E[ X n | G ]  E[ X | G ] with
probability one.
2. Let X and Y be random variables on (, F, P ) and let Z be another random variable,
such that Z = f(X) for some Borel measurable function f : R  R . Show that
E[ E[Y | X ] | Z ]  E[Y | Z ].
3. Let (, F, P) be a probability space. If X is an integrable random variable, Y, is a
bounded random variable, and G  F where G is a  -field, show that
E{YE[ X | G ]}  E{ XE[Y | G ]}
4. Let (, F, P) be a probability space and let X be an integrable random variable such
that E[ X 2 ]   . Show that if G1  G2 , then
E[( X  E ( X | G2 )) 2 ]  E[( X  E ( X | G1 )) 2 ]
i.e. the dispersion of X about its conditional mean becomes smaller as the  -field grows.
5. Let (, F, P) be a probability space with G  F where G is a  -field. Prove that
Var{E[Y | G ]}  Var(Y ). Hint: see problem 34.10(b), page 456 in Billingsley.
Assignment IS Complete