Statistics 620
Midterm takehome exam, Fall 2011
Name:
UMID #:
Midterm Exam
THIS IS THE SAME AS THE IN-CLASS MIDTERM EXAM. IT IS DUE IN CLASS ON
10/27.
YOU MAY CONSULT THE TEXTBOOK AND WIKIPEDIA. PLEASE DO
NOT CONSULT OTHER SOURCES OF ANY KIND, INCLUDING
Homework 7 (Stat 620, Fall 2011)
Due Thu Nov 10, in class
1. Show that a continuous-time Markov chain is regular, given (a) that i < M < for all i or
(b) that the corresponding embedded discrete-time Markov chain with transition probabilities
Pij is irred
Homework 6 (Stat 620, Fall 2011)
Due Thursday Nov 3, in class
1. In a branching process the number of ospring per individual has a Binomial (2, p) distribution. Starting with a single individual, calculate:
(a) the extinction probability;
(b) the probabil
Homework 6 (Stat 620, Fall 2011)
Due Thursday Nov 3, in class
1. In a branching process the number of ospring per individual has a Binomial (2, p) distribution. Starting with a single individual, calculate:
(a) the extinction probability;
(b) the probabil
Homework 5 (Stat 620, Fall 2011)
Due Thursday Oct 13, in class
1. Prove that if the number of state is n, and if state j is accessible from state i, then it is
accessible in n or fewer steps.
k
Solution: j is accessible from i if, for some k 0, Pij > 0. N
Homework 5 (Stat 620, Fall 2011)
Due Thursday Oct 13, in class
1. Prove that if the number of state is n, and if state j is accessible from state i, then it is
accessible in n or fewer steps.
2. For states i, j, k with k = j, let
n
Pij/k = P cfw_Xn = j, X
Homework 4 (Stat 620, Fall 2011)
Due Thu Oct 6, in class
1. Let A(t) and Y (t) denote respectively the age and excess at t. Find:
(a) Pcfw_Y (t) > x|A(t) = s.
(b) Pcfw_Y (t) > x|A(t + x/2) = s.
(c) Pcfw_Y (t) > x|A(t + x) > s for a Poisson process.
(d) Pc
Homework 4 (Stat 620, Fall 2011)
Due Thu Oct 6, in class
1. Let A(t) and Y (t) denote respectively the age and excess at t. Find:
(a) Pcfw_Y (t) > x|A(t) = s.
(b) Pcfw_Y (t) > x|A(t + x/2) = s.
(c) Pcfw_Y (t) > x|A(t + x) > s for a Poisson process.
(d) Pc
Homework 3 (Stat 620, Fall 2011)
Due Thu Sept 29, in class
1. Prove the renewal equation
t
m(t x) dF (x)
m(t) = F (t) +
0
Hint: One approach is to use the identity E[X ] = E E[X |Y ] for appropriate choices of X
and Y .
Solution:
m(t) = E(N (t)
= E(E(N (t
Homework 3 (Stat 620, Fall 2011)
Due Thu Sept 29, in class
1. Prove the renewal equation
t
m(t x) dF (x)
m(t) = F (t) +
0
Hint: One approach is to use the identity E[X ] = E E[X |Y ] for appropriate choices of X
and Y .
2. Prove that the renewal function
Homework 2 (Stat 620, Fall 2011)
Due Thu Sept 22, in class
1. Let cfw_N (t), t 0 be a Poisson process with rate . Calculate E N (t)N (t + s) .
Comment: Please state carefully where you make use of basic properties of Poisson processes,
such as stationary,
Homework 2 (Stat 620, Fall 2011)
Due Thu Sept 22, in class
1. Let cfw_N (t), t 0 be a Poisson process with rate . Calculate E N (t)N (t + s) .
Comment: Please state carefully where you make use of basic properties of Poisson processes,
such as stationary,
Homework 1 (Stat 620, Fall 2011)
Due Thu Sept 15, in class
1. (a) Let N denote a nonnegative integer-valued random variable. Show that
Pcfw_N k =
E[N ] =
k=1
Pcfw_N > k .
k=0
(b) In general show that if X is nonnegative with distribution F , then
E[X ] =
Homework 1 (Stat 620, Fall 2011)
Due Thu Sept 15, in class
1. (a) Let N denote a nonnegative integer-valued random variable. Show that
Pcfw_N k =
E[N ] =
k=1
Pcfw_N > k .
k=0
(b) In general show that if X is nonnegative with distribution F , then
E[X ] =