<< Chapter < Page Chapter >> Page >

The convergence process is clear, and the agreement with the error is close to the predicted. We have not determined the factor a , and we have approximated the long run matrix P 0 with P 16 . This exhibits a practical criterion for sufficient convergence. If the rows of P n agree within acceptable precision, then n is sufficiently large. For example, if we consider agreement to four decimal places sufficient, then

P10 = P^10 P10 =0.2858 0.2847 0.2632 0.1663 0.2858 0.2847 0.2632 0.16630.2858 0.2847 0.2632 0.1663 0.2858 0.2847 0.2632 0.1663

shows that n = 10 is quite sufficient.

Simulation of finite homogeneous markov sequences

In the section, "The Quantile Function" , the quantile function is used with a random number generator to obtain a simple random sample from a given population distribution. In thissection, we adapt that procedure to the problem of simulating a trajectory for a homogeneous Markov sequences with finite state space.

    Elements and terminology

  1. States and state numbers . We suppose there are m states, usually carrying a numerical value. For purposes of analysis and simulation, we number the states 1 through m . Computation is carried out with state numbers; if desired, these can be translated intothe actual state values after computation is completed.
  2. Stages, transitions, period numbers, trajectories and time . We use the term stage and period interchangeably. It is customary to number the periods or stages beginning withzero for the initial stage. The period number is the number of transitions to reach that stage from the initial one. Zero transitions are required to reach the original stage (periodzero), one transition to reach the next (period one), two transitions to reach period two, etc. We call the sequence of states encountered as the system evolves a trajectory or a chain . The terms “sample path” or “realization of the process” are also used in the literature. Now if the periods are of equal time length, the number of transitions is ameasure of the elapsed time since the chain originated. We find it convenient to refer to time in this fashion. At time k the chain has reached the period numbered k . The trajectory is k + 1 stages long, so time or period number is one less than the number of stages.
  3. The transition matrix and the transition distributions . For each state, there is a conditional transition probability distribution for the next state. These arearranged in a transition matrix . The i th row consists of the transition distribution for selecting the next-period state when the current state number is i . The transition matrix P thus has nonnegative elements, with each row summing to one. Such a matrix is known as a stochastic matrix .

    The fundamental simulation strategy

  1. A fundamental strategy for sampling from a given population distribution is developed in the unit on the Quantile Function. If Q is the quantile function for the population distribution and U is a random variable distributed uniformly on the interval [ 0 , 1 ] , then X = Q ( U ) has the desired distribution. To obtain a sample from the uniform distribution usea random number generator. This sample is “transformed” by the quantile function into a sample from the desired distribution.
  2. For a homogeneous chain, if we are in state k , we have a distribution for selecting the next state. If we use the quantile function for that distribution anda number produced by a random number generator, we make a selection of the next state based on that distribution. A succession of these choices, with the selectionof the next state made in each case from the distribution for the current state, constitutes a valid simulation of a trajectory.

Questions & Answers

how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
Idrissa Reply
im all ears I need to learn
right! what he said ⤴⤴⤴
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
how do you translate this in Algebraic Expressions
linda Reply
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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
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 the best Algebra and trigonometry course in your pocket!

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?