<< Chapter < Page Chapter >> Page >

The transition matrix P for a homogeneous Markov chain is as follows (in m-file npr16_08.m ):

P = 0 . 2 0 . 5 0 . 3 0 0 0 0 0 . 6 0 . 1 0 . 3 0 0 0 0 0 . 2 0 . 7 0 . 1 0 0 0 0 0 0 0 0 . 6 0 . 4 0 0 0 0 0 0 . 5 0 . 5 0 0 0 . 1 0 . 3 0 0 . 2 0 . 1 0 . 1 0 . 2 0 . 1 0 . 2 0 . 1 0 . 2 0 . 2 0 . 2 0
  1. Note that the chain has two subchains, with states { 1 , 2 , 3 } and { 4 , 5 } . Draw a transition diagram to display the two separate chains. Can any state in one subchain be reached from any state in the other?
  2. Check the convergence as in part (a) of [link] . What happens to the state probabilities for states 6 and 7 in the long run? What does thatsignify for these states? Can these states be reached from any state in either of the subchains? How would you classify these states?

Increasing power P n show the probability of being in states 6, 7 go to zero. These states cannot be reached from any of the other states.

Got questions? Get instant answers now!

The transition matrix P for a homogeneous Markov chain is as follows (in m-file npr16_09.m ):

P = 0 . 1 0 . 2 0 . 1 0 . 3 0 . 2 0 0 . 1 0 0 . 6 0 0 0 0 0 . 4 0 0 0 . 2 0 . 5 0 0 . 3 0 0 0 0 . 6 0 . 1 0 0 . 3 0 0 . 2 0 . 2 0 . 1 0 . 2 0 0 . 1 0 . 2 0 0 0 . 2 0 . 7 0 0 . 1 0 0 0 . 5 0 0 0 0 0 . 5
  1. Check the transition matrix P for convergence, as in part (a) of [link] . How many steps does it take to reach convergence to four or more decimal places? Does this agree with the theoretical result?
  2. Examine the long run transition matrix. Identify transient states.
  3. The convergence does not make all rows the same. Note, however, that there are two subgroups of similar rows. Rearrange rows and columns in the longrun Matrix so that identical rows are grouped. This suggests subchains. Rearrange the rows and columns in the transition matrix P and see that this gives a pattern similar to that for the matrix in [link] . Raise the rearranged transition matrix to the power for convergence.

Examination of P 16 suggests sets { 2 , 7 } and { 3 , 4 , 6 } of states form subchains. Rearrangement of P may be done as follows:

PA = P([2 7 3 4 6 1 5], [2 7 3 4 6 1 5]) PA =0.6000 0.4000 0 0 0 0 0 0.5000 0.5000 0 0 0 0 00 0 0.2000 0.5000 0.3000 0 0 0 0 0.6000 0.1000 0.3000 0 00 0 0.2000 0.7000 0.1000 0 0 0.2000 0.1000 0.1000 0.3000 0 0.1000 0.20000.2000 0.2000 0.1000 0.2000 0.1000 0.2000 0 PA16 = PA^16PA16 = 0.5556 0.4444 0 0 0 0 00.5556 0.4444 0 0 0 0 0 0 0 0.3571 0.3929 0.2500 0 00 0 0.3571 0.3929 0.2500 0 0 0 0 0.3571 0.3929 0.2500 0 00.2455 0.1964 0.1993 0.2193 0.1395 0.0000 0.0000 0.2713 0.2171 0.1827 0.2010 0.1279 0.0000 0.0000

It is clear that original states 1 and 5 are transient.

Got questions? Get instant answers now!

Use the m-procedure inventory1 (in m-file inventory1.m ) to obtain the transition matrix for maximum stock M = 8, reorder point m = 3 , and demand D Poisson(4).

  1. Suppose initial stock is six. What will the distribution for X n , n = 1 , 3 , 5 (i.e., the stock at the end of periods 1, 3, 5, before restocking)?
  2. What will the long run distribution be?
inventory1 Enter value M of maximum stock 8Enter value m of reorder point 3 Enter row vector of demand values 0:20Enter demand probabilities ipoisson(4,0:20) Result is in matrix Pp0 = [0 0 0 0 0 0 1 0 0];p1 = p0*P p1 =Columns 1 through 7 0.2149 0.1563 0.1954 0.1954 0.1465 0.0733 0.0183Columns 8 through 9 0 0p3 = p0*P^3 p3 =Columns 1 through 7 0.2494 0.1115 0.1258 0.1338 0.1331 0.1165 0.0812Columns 8 through 9 0.0391 0.0096p5 = p0*P^5 p5 =Columns 1 through 7 0.2598 0.1124 0.1246 0.1311 0.1300 0.1142 0.0799Columns 8 through 9 0.0386 0.0095a = abs(eig(P))' a =Columns 1 through 7 1.0000 0.4427 0.1979 0.0284 0.0058 0.0005 0.0000Columns 8 through 9 0.0000 0.0000a(2)^16 ans =2.1759e-06 % Convergence to at least five decimals for P^16 pinf = p0*P^16 % Use arbitrary p0, pinf approx p0*P^16pinf = Columns 1 through 7 0.2622 0.1132 0.1251 0.1310 0.1292 0.1130 0.0789Columns 8 through 9 0.0380 0.0093
Got questions? Get instant answers now!

Questions & Answers

show that the set of all natural number form semi group under the composition of addition
Nikhil Reply
what is the meaning
Dominic
explain and give four Example hyperbolic function
Lukman Reply
_3_2_1
felecia
⅗ ⅔½
felecia
_½+⅔-¾
felecia
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
SABAL Reply
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
Pawel
ok
Ifeanyi
on number 2 question How did you got 2x +2
Ifeanyi
combine like terms. x + x + 2 is same as 2x + 2
Pawel
x*x=2
felecia
2+2x=
felecia
×/×+9+6/1
Debbie
Q2 x+(x+2)+(x+4)=60 3x+6=60 3x+6-6=60-6 3x=54 3x/3=54/3 x=18 :. The numbers are 18,20 and 22
Naagmenkoma
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
mariel Reply
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
Pawel
how do I set up the problem?
Harshika Reply
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
hello, I am happy to help!
Shirley Reply
please can go further on polynomials quadratic
Abdullahi
hi mam
Mark
I need quadratic equation link to Alpa Beta
Abdullahi Reply
find the value of 2x=32
Felix Reply
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
yes i wantt to review
Mark
16
Makan
x=16
Makan
use the y -intercept and slope to sketch the graph of the equation y=6x
Only Reply
how do we prove the quadratic formular
Seidu Reply
please help me prove quadratic formula
Darius
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
Shirley Reply
thank you help me with how to prove the quadratic equation
Seidu
may God blessed u for that. Please I want u to help me in sets.
Opoku
what is math number
Tric Reply
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
can you teacch how to solve that🙏
Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive
Samson Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Applied probability' conversation and receive update notifications?

Ask