# 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

what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
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