Stat 8273, Spring 2016
Homework 2
Due Tue, Feb/16, at the beginning of class
1. The goal for this problem is to prove the following result from class:
Theorem. Suppose that for each n N, the random variables Yn,j , j = 1, ., n,
are independent Bernoulli w
Stat 8273, Spring 2014
Homework 5
Due Tue, Mar 8, at the beginning of class
Suppose we have a sequence of random variables (Xn ) with limit X := limn Xn .
Often it is easy to calculate the expectations E[Xn ], and it is then tempting to conclude
that
h
i
Stat 8273, Spring 2016
Homework 6
Due Tue, Apr 12, at the beginning of class
1. Consider an ordinary arithmetic renewal process with span = 1, inter-arrival
times Xi 1, and := E[X1 ] < . Let R(m) := SN (m)+1 m be the residual
lifetime at time m N0 . For f
Stat 8273, Spring 2016
Homework 3
Due Tue, Feb 23, at the beginning of class
Review section 1.5 in our textbook on conditional expectation.
1. Question 2.4 on page 89 in our textbook by Ross.
Hint: Condition on everything we know from observing the proces
Stat 8273, Spring 2016
Homework 4
Due Tue, Mar 1, at the beginning of class
1. Suppose (Nt : t > 0) and (Mt : t > 0) are independent Poisson counting processes
with rates and , respectively. Let := infcfw_t > 0 : Nt = 1 be the time of the
first occurrence
Stat 8273, Spring 2016
Homework 7
Due Tue, Apr 19, at the beginning of class
1. Is p q random walk on Z is irreducible? Prove your answer.
P
2. Let Sn := ni=1 Xi be the event times of an ordinary arithmetic renewal process
with span (period) one, and let
Stat 8273, Spring 2016
Homework 8
Due Tue, Apr 26
1. The goal in this problem is to prove that simple random walk on the integers
is null recurrent. (We proved in class that it is recurrent.) Note that this
means that the process is guaranteed to return t
Stat 8273, Fall 2016
Homework 1
Due Tue, Jan 26, at the beginning of class
Definition. Let f, g be two real valued functions. We say that f (x) is (in) Big-Oh of
g(x), denoted by f (x) = O g(x) , as x a [, ], if there exists a constant M
and a neighborhoo