Reference no: EM13909255
Assume a friend has developed an excellent program for finding the steady-state probabilities for finite-state Markov chains. More precisely, given the transition matrix [P], the program returns limn Pn for each i. Assume all chains are aperiodic.
(a) You want to find the expected time to first reach a given state k starting from a different state m for a Markov chain with transition matrix [P]. You modify the matrix to [P∗] where P∗ = 1, P∗ = 0 for j /= m, and P∗ = Pij otherwise. How do you find the desired first-passage time from the program output given [P∗] as an input? Hint: The times at which a Markov chain enters any given state can be considered as renewals in a (perhaps delayed) renewal process.
(b) Using the same [P∗] as the program input, how can you find the expected number of returns to state m before the first passage to state k?
(c) Suppose, for the same Markov chain [P] and the same starting state m, you want to find the probability of reaching some given state n before the first passage to k. Modify [P] to some [P∗∗] so that the above program withP∗∗ as an input allows you to easily find the desired probability.
(d) Let Pr{X(0) = i} = Qi, 1 ≤ i ≤ M be an arbitrary set of initial probabilities for the same Markov chain [P] as above. Show how to modify [P] to some [P∗∗∗] for which the steady-state probabilities allow you to easily find the expected time of the first passage to state k.
Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.
|
How many ways can she stack the 5 books she must carry
: How many license plates using 6 digits can the state of Vermont issue if repetition of digits is permitted - how many ways can the horses finish the race - How many of the first 100 natural numbers are multiples of 2 or multiples of 5?
|
|
Show that xn and ij are statistically independent
: Show that Xn and IJ≥n are statistically independent. Give the simplest characterization you can of the event {J ≥n}. Show that Xn and IJ>n are statistically dependent.
|
|
Write a marketing plan proposal for the social issue
: Write a marketing plan proposal for the social issue that was the subject of the Identification of a Social Cause Memo assignment. Identify a social cause that would benefit from a social marketing campaign.
|
|
Physician that he has pleurisy
: John has been suffering agonizing pain with each breath and has been informed by the physician that he has pleurisy.
|
|
How can you find the expected number of returns to state m
: Using the same [P∗] as the program input, how can you find the expected number of returns to state m before the first passage to state k?
|
|
Develop a marketing plan for a trophy and engraving shop
: Develop a Marketing Plan for a Trophy & Engraving shop. Develop the situational analysis, executive summary, goals and objectives, advantages, strategic focus, the market analysis, implementation, evaluation and control, competition analysis and S..
|
|
Cultivating single species on a plot be so productive
: If rich species communities are more productive than species poor coomunites how can modern agriculture which is based almost entirely on cultivating single species on a plot be so productive?
|
|
What type of personality does the brand have
: Select a brand that most are not so familiar with. What type of personality does the brand have. What are the key product features or qualities that define the brand?
|
|
Medication to the surface of the skin
: Many medications can be administered transdermally by applying patches that contain the medication to the surface of the skin. These patches can be attached anywhere on the skin except the palms of the hands and the soles of the feet. Why?
|