Long Question: Markov chain - Probability
Long Question: Markov chain In many applications, some variables evolve over time in a random (stochastic) way in that even if you know everything up to time t, it is still impossible to know for sure which values the variables take in the future. Stochastic processes are mathematical models that describe these phenomena when the randomness is driven by something out of the control of the relevant decision makers. For example, the stock prices can be (and usually are) modeled as stochastic processes, since it is difficult for an investor to affect them. However, in a chess game, the uncertainty in your opponents’ future moves are not modeled as stochastic processes as they are made by a decision maker with the goal to defeat you and they may be adjusted based on your moves. One simple and widely used class of stochastic processes are Markov chains. In this question, we study Markov chains on a finite state space. There is a sequence of random variables x0, x1, ..., xt, ..., each taking value in a finite set S called the state space. The subscripts have the interpretation of time. Therefore given an integer time t, x0, ..., xt are assumed to be known at the time, while xt+1, xt+2, ... remain random. For convenience, we label states with positive integers: S = {1, ..., n}, where n is the number of possible states. (a) Is S a good choice of sample space for describing this stochastic process? No matter what the answer is, for the rest of the question we assume that a good sample space Ω (which might be S if your answer is “yes”) has been chosen to carry the random variables. (b) The fundamental assumption of Markov chains is that given x0, ..., xt, the probability that xt+1 = j is pji where i is the value of xt. This holds for every t = 0, 1, ... and the numbers pji are independent of t. More precisely, for any fixed t and the event A that x0 = a0, ..., xt = at (where a0, a1, ..., at are integers in S), P(x−1t+1(j) ∩ A) = P(A)pj,at; this holds for all a0, a1, ..., at. (The so-called Markov property means that the value of xt+1 only depends on xt and not on history further back.) Let P be the n × n matrix whose (j, i) entry is pji for all i, j ∈ S; it is called the transition matrix of the Markov chain. Show that for the probabilities to be well-defined for all laws of x0, each column of P sums up to one. (c) A law on the state space S can be represented by an n × 1 matrix (column vector), whose (i, 1) entry is the probability of {i}. A function f : S → R can be represented by a 1 × n matrix whose (1, i) entry is the number f(i). (Notice that f is not a random variable unless your answer to Part (a) is “yes”.) If the law of x0 is v0, then what is the interpretation of fv0 (matrix product)? (d) In what follows, fix a law v0 of x0 and a function f : S → R. What is the law of x1? (e) Notice that fP is a 1×n matrix, so it represents a function on S. What does that function mean intuitively? (As we can see, the transformation from f to fP is another way to describe the transition matrix P , and this transformation plays an important role when studying more complicated Markov processes.) (f) What is the law of xt? 3
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Your assignment may be more than 5 paragraphs but not less. INSTRUCTIONS:  To access the FNU Online Library for journals and articles you can go the FNU library link here:  https://www.fnu.edu/library/ In order to n that draws upon the theoretical reading to explain and contextualize the design choices. Be sure to directly quote or paraphrase the reading ce to the vaccine. Your campaign must educate and inform the audience on the benefits but also create for safe and open dialogue. A key metric of your campaign will be the direct increase in numbers.  Key outcomes: The approach that you take must be clear Mechanical Engineering Organic chemistry Geometry nment Topic You will need to pick one topic for your project (5 pts) Literature search You will need to perform a literature search for your topic Geophysics you been involved with a company doing a redesign of business processes Communication on Customer Relations. 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Furman was originally sentenced to death because of a murder he committed in Georgia but the court debated whether or not this was a violation of his 8th amend One of the first conflicts that would need to be investigated would be whether the human service professional followed the responsibility to client ethical standard.  While developing a relationship with client it is important to clarify that if danger or Ethical behavior is a critical topic in the workplace because the impact of it can make or break a business No matter which type of health care organization With a direct sale During the pandemic Computers are being used to monitor the spread of outbreaks in different areas of the world and with this record 3. Furman v. Georgia is a U.S Supreme Court case that resolves around the Eighth Amendments ban on cruel and unsual punishment in death penalty cases. The Furman v. Georgia case was based on Furman being convicted of murder in Georgia. 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