Name_
June 29, 2014
Copyrighted Jeffrey A Beckley
STAT 490
Fall 2012
Test 2
October 30, 2012
1. Datsenka Dog Insurance Company has developed the following mortality table for dogs:
Age
0
1
2
3
4
lx
2000
1950
1850
1600
1400
Age
5
6
7
8
9
lx
1200
1000
700
3
Homework 7 - Additional Problems
7.1. There is a board game for kids called Hi-ho Cherry-O that has the following rules.
Every player has a tree with 10 cherries on it and an empty bucket. Each turn the player
spins a spinner which determines how many che
Homework 6 - Additional Problems
6.1. Let Xn be a Markov chain on Z+ = cfw_0, 1, 2, . . . with transition probabilities given by
p(0, j) = (1 )j for all j 0 and p(i, i 1) = 1 for all i 1.
That is, when the Markov chain is at 0 it jumps a distance to the r
Homework 5 - Additional Problems
5.1. In class we showed that if the transition probability matrix p is doubly stochastic then
the uniform ditribution on the state space I was a stationary distribution.
Is the converse true? That is, if the uniform distri
Homework 4 - Additional Problems
4.1. Let Xn be the Ehrenfest Markov chain with N balls.
a) Prove that
1
n n
n
lim
Xk =
k=1
b) Explain why it is NOT true that limn Xn =
N
.
2
N
.
2
4.2. Suppose that a Markov chain on a nite state space is irreducible and
Homework 2 - Additional Problems
2.1. Let cfw_Xn n0 be the gamblers ruin Markov chain with p = .45 and N = 6 (compare
with Example 1.1 in the book). That is, if you currently have i > 0 dollars, in the next
round the probability of winning $1 is p(i, i +
Homework 2 - Additional Problems
2.1. Let cfw_Xn n0 be the gamblers ruin Markov chain with p = .45 and N = 6 (compare
with Example 1.1 in the book). That is, if you currently have i > 0 dollars, in the next
round the probability of winning $1 is p(i, i +
Homework 3 - Additional Problems
3.1. For each of the following transition matrices, determine if the Markov chain is irreducible. Justify your answer (it may help to give a graphical representation of the Markov
chain).
0
1/3
1/3
a)
0
0
0
1/2
0
b) 1/
Homework 1 - Additional Problems
1.1. Solve the following matrix equations.
a)
2 3
1 4
x
y
=
1
1
b) Find all solutions to
x y
1.2. Let A =
1/3 2/3
1/4 3/4
= x y
4 8
. Find a diagonal matrix D and an invertible matrix S such that
3 9
A = SDS 1 .
1.3. Suppo
Homework 3 - Additional Problems
3.1. For each of the following transition matrices, determine if the Markov chain is irreducible. Justify your answer (it may help to give a graphical representation of the Markov
chain).
0 1/2 1/2 0
0
0
1/3 0 2/3 0
0
0
1
Homework 1 - Additional Problems
1.1. Solve the following matrix equations.
a)
2 3
1 4
x
y
Answer:
=
x
y
=
1
1
7/11
1/11
b) Find all solutions to
x y
1/3 2/3
1/4 3/4
= x y
Answer: All solutions are of the form x y = 3t/8 t .
1.2. Let A =
4 8
. Find a diag
STAT 490
Test 3
Fall 2012
December 11, 2012
1. You are given:
a. 1000 A20 200
b. 1000 A35 360
c.
1000 A20:35 440
d.
v 0.9
Calculate the annual net benefit premium for a fully discrete last survivor whole life issued to
(20) and (35) which has a death bene
STAT 490
Fall 2012
Test 2
October 30, 2012
1. Datsenka Dog Insurance Company has developed the following mortality table for dogs:
Age
0
1
2
3
4
lx
2000
1950
1850
1600
1400
Age
5
6
7
8
9
lx
1200
1000
700
300
0
Datsenka sells a whole life annuity based on
Homework 4 - Additional Problems
4.1. Let Xn be the Ehrenfest Markov chain with N balls.
a) Prove that
n
1X
N
Xk = .
n n
2
k=1
lim
Answer: Since the Ehrenfest chain is an irreducible Markov chain with finite state
space, we know that the limit exists and