Math 544, Assignment 1
Due 2013-09-17, 11:00
Problem 1. Prove the inclusion-exclusion principle. Bonus: Show that partial sums are
alternate upper and lower bounds for P(Ai ).
Problem 2. A -algebra F is said to be generated by a partition if there is some
MATH 418/544, Assignment 7 solutions
1
Problem 1. If S, T are stopping times and m > 0 is xed, determine which
of the following are stopping times: T m, T m, S T , S T , T + m,
T + S, T m, T + m.
Solution. Only T m is not a stopping time. In all other cas
MATH 544, Assignment 8 solutions
1
Problem 1. Consider a random walk on Z with increments Xi that are some
bounded random variable. Suppose P(X > 0) and P(X < 0) are non-zero, and
EX = 0. Show that there is some 0 < q = 1 so that q Sn is a martingale.
Sol
MATH 544, Assignment 7 solutions
1
Problem 1. If S, T are stopping times and m > 0 is xed, determine which
of the following are stopping times: T m, T m, S T , S T , T + m,
T + S, T m, T + m.
Solution. Only T m is not a stopping time. In all other cases t
MATH 544, Assignment 7
Due 2014-11-26, 10:00
Problem 1. If S, T are stopping times and m > 0 is xed, determine which
of the following are stopping times: T m, T m, S T , S T , T + m,
T + S, T m, T + m.
Problem 2. Fill in details to prove that the random w
MATH 544, Assignment 8
Due 2014-12-02
Problem 1. Consider a random walk on Z with increments Xi that are some
bounded random variable. Suppose P(X > 0) and P(X < 0) are non-zero, and
EX = 0. Show that there is some 0 < q = 1 so that q Sn is a martingale.
MATH 544, Assignment 6 solutions
1
Problem 1. Prove that X is periodic if and only if X takes values in aZ
(multiples of a) for some a.
solution. If (t) = (0) = 1, then EeitX = 1. This can only happen if
eitX = 1 always (since |eitX | = 1). Thus X = 2n/t
Math 544, Assignment 4
Due 2013-10-22, 11:00
Problem 1. For given p, construct a tree where pc = p and (pc ) > 0.
Problem 2. Find variables Xn , all with mean 0, variance 1 and P(|Xn | > 0.1) 0.
Problem 3. Give an example of uncorrelated random variables
MATH 544, Assignment 5
Due 2014-11-05, 10:00
Problem 1. a. If Xn X and Yn Y in probability show that Xn + Yn
X + Y in probability.
b. Do the same for convergence in L1 . c. Show this fails for convergence in
distribution.
prob
prob
n
n
Problem 2. If Xn X
Math 544, Assignment 3
Due 2013-10-08, 11:00
Denition 1. The m-ary tree is a graph with a single vertex on level 0, and each vertex on
level k has m edges to vertices on level k + 1, all distinct (so there are mk vertices on level k.
Problem 1. Consider p
MATH 544, Assignment 6
Due 2014-11-12, 10:00
Problem 1. Prove that X is periodic if and only if X takes values in aZ
(multiples of a) for some a.
Problem 2. Use characteristic functions to prove the weak law of large
numbers.
c
Problem 3. A The Cauchy ran
Math 544, Assignment 2
Due 2013-10-01, 11:00
Problem 1. nd n events so that any n 1 are independent but all together are not.
Problem 2. In the previous problem, show that | 2n1
Problem 3. a. Let (Ai )in be independent events with probability 1/n each. Fi
Math 544, Assignment 4 solutions
Problem 1. For given p, construct a tree where pc = p and (pc ) > 0.
Consider a symmetric tree, where a vetex at level k has dk children, so that level n has
volume Vn = i<n di . Let Xn be the number of vertices in level n
MATH 418/544, Assignment 8 solutions
1
Problem 1. Consider a random walk on Z with increments Xi that are some
bounded random variable. Suppose P(X > 0) and P(X < 0) are non-zero, and
EX = 0. Show that there is some 0 < q = 1 so that q Sn is a martingale.
Math 544, Assignment 1 solutions
Problem 1. Prove the inclusion-exclusion principle. Bonus: Show that partial sums are
alternate upper and lower bounds for P(Ai ).
We write the principle as
(1)|S| P(iS Ai ) = 1
P(Ac ) =
i
P(Ai ) + . . . .
S
Let BI = iI A
MATH 544, Assignment 5
Due 2014-11-05, 10:00
Problem 1. a. If Xn X and Yn Y in probability show that Xn + Yn
X + Y in probability.
b. Do the same for convergence in L1 . c. Show this fails for convergence in
distribution.
solution. a. with high probabili
Math 544, Assignment 3
Due 2013-10-08, 11:00
Denition 1. The m-ary tree is a graph with a single vertex on level 0, and each vertex on
level k has m edges to vertices on level k + 1, all distinct (so there are mk vertices on level k.
Problem 1. Consider p
MATH 418/544, Assignment 6 solutions
1
Problem 1. Prove that X is periodic if and only if X takes values in aZ
(multiples of a) for some a.
solution. If (t) = (0) = 1, then EeitX = 1. This can only happen if
eitX = 1 always (since |eitX | = 1). Thus X = 2
Math 544, Assignment 2 solutions
Problem 1. nd n events so that any n 1 are independent but all together are not.
Consider n 1 independent fair coins, and let the events be that coin i is heads, and
that the total number of heads is odd. It is easy to ver
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