# 16.2 Problems on conditional independence, given a random vector  (Page 2/2)

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

$P=\left[\begin{array}{ccccccc}0.2& 0.5& 0.3& 0& 0& 0& 0\\ 0.6& 0.1& 0.3& 0& 0& 0& 0\\ 0.2& 0.7& 0.1& 0& 0& 0& 0\\ 0& 0& 0& 0.6& 0.4& 0& 0\\ 0& 0& 0& 0.5& 0.5& 0& 0\\ 0.1& 0.3& 0& 0.2& 0.1& 0.1& 0.2\\ 0.1& 0.2& 0.1& 0.2& 0.2& 0.2& 0\end{array}\right]$
1. Note that the chain has two subchains, with states $\left\{1,2,3\right\}$ and $\left\{4,5\right\}$ . 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.

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

$P=\left[\begin{array}{ccccccc}0.1& 0.2& 0.1& 0.3& 0.2& 0& 0.1\\ 0& 0.6& 0& 0& 0& 0& 0.4\\ 0& 0& 0.2& 0.5& 0& 0.3& 0\\ 0& 0& 0.6& 0.1& 0& 0.3& 0\\ 0.2& 0.2& 0.1& 0.2& 0& 0.1& 0.2\\ 0& 0& 0.2& 0.7& 0& 0.1& 0\\ 0& 0.5& 0& 0& 0& 0& 0.5\end{array}\right]$
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 $\left\{2,7\right\}$ and $\left\{3,4,6\right\}$ 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.

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 $D\sim$ 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

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive