Maths - Question and Answer

MATHS

Table of Contents
Question 1................................................................................................................ 2
a)......................................................................................................................... 2
b)........................................................................................................................ 3
Question 2................................................................................................................ 3
Question 3................................................................................................................ 4
a)......................................................................................................................... 4
b)........................................................................................................................ 5
c)......................................................................................................................... 5
Question 4................................................................................................................ 5
a)......................................................................................................................... 5
b)........................................................................................................................ 6
Question 5................................................................................................................ 6
i)......................................................................................................................... 6
ii)........................................................................................................................ 6
Question 6................................................................................................................ 7
a)......................................................................................................................... 7
b)........................................................................................................................ 7
i)...................................................................................................................... 7
ii)..................................................................................................................... 8
c)......................................................................................................................... 9
i)...................................................................................................................... 9
ii)..................................................................................................................... 9
iii).................................................................................................................... 9
iv)................................................................................................................... 10
Reference............................................................................................................... 11

Question 1
a)
Let as consider the x=(xn :nε N0)
The homogenous markov chain on the state space is S= {-1, 0, 1}
( 0 1 /2 1/2
1/2 0 1/2
1/2 1 /2 0 )

=
( 0 0.5 0.5
0.5 0 0.5
0.5 0.5 0 ) (−1 0 1 )
( 0∗−1 0.5∗0 0.5∗1
0.5∗−1 0∗0 0.5∗1
0.5∗1 0.5∗0 0∗1 )=
( 0 0 0.5
−0.5 0 0.5
0.5 0 0 )
A zero element in the transition matrix indicate that the transition is not possible.
To calculating the probability transition matrix is,
E ( X 1
X 0 ) =b and E ( X 2
X 0 ) =b
E ( X 1
X 0 ) =0.5, E ( X 2
X 0 ) =0.5
b)
P(X 0=−1 ¿=¿ P( X 0=0 ¿= P(X 0=1¿
=
( 0 0.5 0.5
0.5 0 0.5
0.5 0.5 0 ) (−1 0 1 )
=
( 0 0 0.5
−0.5 0 0.5
0.5 0 0 )
E(x1)=0 and E(x2)=0.25
Question 2
S = {0, 1, 2}
=
(1/3 1/6 1/2
1/ 4 1/3 5/12
1/2 1 /4 1/ 4 )=
( 0.33 0.166 0.5
0.25 0.33 0.416
0.5 0.25 0.25 )
The stationary distribution is ( π1, π2 , π3)
πP=π