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 probability measure
given by
P (cfw_1) =
1
1
1
1
1
1
, P (cfw_
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 , .
,
ANS: We have seen en-route to (5.2.2) that , < almost
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 characteristic functions are
Sbk () = [cos(/ k)]k .
ANS: Let Xi
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 , Gt ) is a MG for Gt = (Ms , s t) and the non-random, non
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 hence the inequality holds.(no need for algebra)
2. Exerc
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 various parameters the extinction is certain.
(a) 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 indicator functions g(x) = IB (x) for B B.
ANS: Let B B be an
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
properties of IP.
(a) Monotonicity. If A B then IP(A) I
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 of a balanced six-sided dice(including their order). C
Stochastic Processes
Stat219/Math136
Fall 2010
Place: Building 320, Room 105
Time: Monday, Wednesday, Friday, 12:15am 1:05am
Instructor:
Jessica Z
un
iga, jzuniga@math.stanford.edu, Building 380 Room 382K
Teaching Assistants:
Philip Labo
Michael Lim
plabo
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 and that a.s. E[Xl |Fn ] = Xn for all l > n.
ANS: Its eas