Do all problems and give the process of your solution. 1. A Markov chain cfw_X0 , X1 , X2 , , has the transition probability 0 0 0.1 1 0.2 0.3 1 2
2 Determine the conditional probabilities (1). P [X1 = 1, X2 = 1|X0 = 0] Solution.
0.3 0.4
0.1 0.8 0.2 0.6
MATH424 HOMEWORK ANSWER
5
Homework 7.
Exercise 1.3
Solution: Since X and Y are independent Poisson distributed random
variable with parameter and , we know X + Y is also a Poisson random
variable with parameter + (See theorem 1.1 in page 268). Therefore,
MATH424 HOMEWORK ANSWER
Homework 6.
Exercise 3.2
Solution: By drawing the transition graph, we will see there are 5 communicating classes: cfw_0, cfw_1, cfw_2, 4, cfw_3 and cfw_5.
)
For state 0, 1 P(n0 = 1 ( 1 )n = 1 < , therefore state 0 is transient.
n=
MATH424 HOMEWORK ANSWER
Homework 9. (Chapter 6)
Exercise 1.1
Solution:
Method 1. We have
P0 (t) = e0 t = et
t
1 t
P1 (t) = 0 e
e1 x P0 ( x)d x
0
t
= e3t
e2 x d x
0
=
1 t 1 3t
ee
2
2
t
P2 (t) = 1 e2 t
e2 x P1 ( x)d x
0
t
1 x 1 x
e e dx
2
2
0
1
1
= 3 et + e
Test #2
Math 424
Spring 2010
Name:
Do all problems and give the process of your solution. 1. (25 points) A Markov chain cfw_X0 , X1 , X2 , , has the transition probability 0
1 0 2 1 1 2
1
1 2
2 0
1 2
0 0
2
1
0
Find the limit
n
lim Pi,j
cfw_n
i, j = 0,