STAT 219/ MATH 136 Midterm Practice Problems
These are the problems from last years midterm.
Problem 1
Consider the probability space (, F, P ) where = cfw_1, 2, 3, 4, 5, 6, F = 2 and P is the probabi
Stat 219 - Stochastic Processes
Homework Set 8, Fall 2010
1. Exercise 5.2.5. Show that E(, ) = by applying Doobs optional stopping theorem for the uniformly
2
integrable stopped martingale Wt
t , .
,
Math 136 - Stochastic Processes
Homework Set 5, Fall 2010
1. Exercise 3.2.21. Consider the random variables Sbk of Example 1.4.13.
(a) Applying Proposition 3.2.6 verify that the corresponding characte
Math 136 - Stochastic Processes
Homework Set 7, Fall 2010
1. Exercise 4.2.4
(a) Deduce from the identity (4.2.1) that if the MG cfw_Mt , t 0 of Proposition 4.2.3 is square-integrable,
then (Mt2 At , G
Math 136 - Stochastic Processes
Homework Set 4, Fall 2010
1. Exercise 2.3.8. The left hand side is the smallest distance from G2 , while by embeddedness, right hand
side is a distance from G2 , and he
STATS 219 - Stochastic Processes
HW 10, Fall 2010
1. Exercise 4.6.9. Let cfw_Zn be a branching process with Z0 = 1. Compute pex in each of the following
situations and specify for which values of the
Stat 219 - Stochastic Processes
Homework Set 3, Fall 2010, Due: October 13
1. Exercise 1.4.31. Prove Proposition 1.4.3 using the following steps.
(a) Verify that the identity (1.4.1) holds for indicat
Stat219 / Math 136 - Stochastic Processes
Homework Set 1, Fall 2010. Due: Wednesday, September 29
1. Exercise 1.1.3. Let (, F, IP) be a probability space and A, B, Ai events in F. Prove the following
Stat 219 - Stochastic Processes
Homework Set 2, Fall 2010, Due: October 6
1. Exercise 1.2.22. Write (, F, P ) for a random experiment whose outcome is a recording of the results of
n independent rolls
Math 136 - Stochastic Processes
Homework Set 6, Fall 2010
1. Exercise 4.1.5. Suppose (Xn , Fn ) is a martingale. Show that then cfw_Xn is also a martingale with respect
to its canonical filtration an