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Mar
21
awarded  Popular Question
Mar
9
awarded  Nice Question
Jan
19
awarded  Yearling
Dec
20
comment Understanding the random variable definition of Markov chains
FWIW I found some lecture notes that answer this question in the fashion I had envisioned (hamilton.ie/ollie/Downloads/ProbMain.pdf) -- see section 3.1.
Dec
20
revised Understanding the random variable definition of Markov chains
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Dec
20
comment Understanding the random variable definition of Markov chains
Well thanks for all your input -- much appreciated.
Dec
20
comment Understanding the random variable definition of Markov chains
Isn't it also valid to think of the chain as a dynamical system exploring a space of states (not necessarily pmf's) with a dynamical rule that is probabilistic so that at any given time, the state is determined by a pmf associated with that time, but is not the pmf itself? This (in addition so my impressions from some literature) is my motivation for wanting to associate the state of a possible trajectory of the chain at each time $k$ with a possible value in the range of a random variable.
Dec
19
comment Understanding the random variable definition of Markov chains
(contd.) sense of a simple way of constructing an appropriate $\Omega$ by thinking of the chain as a probabilistic experiment in which one generates a sequence of states (elements of the ranges of the random variables), and in that sense an outcome of the experiment would be such a sequence, leading to the sample space being the set of all such sequences. Imagine flipping one of two biased coins depending on the outcome of the last flip, then I want to try to take $\Omega$ to be the set of infinite sequences of $H$ or $T$, and then $X_k$ is $1$ if the $k$th flip is heads, and $0$ otherwise.
Dec
19
comment Understanding the random variable definition of Markov chains
Thanks, but I don't think this answers the question. I know that in a practical sense, knowing $\Omega$ isn't important, but I still want to know if the way I attempted to construct $\Omega$ works. I understand why one might want to view the trajectory of the chain as a sequence of pmf's, but is that necessarily the way it's commonly viewed? If so, why would the range of the random variables often (as far as I can tell) be called the "state space" of the chain? Lastly, I wasn't thinking of $X(s_0, s_1\dots)$ as the probability of various sequences of states, I was trying to make
Dec
19
revised Understanding the random variable definition of Markov chains
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Dec
19
revised Understanding the random variable definition of Markov chains
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Dec
19
comment Understanding the random variable definition of Markov chains
@A.S. Thank you. I will update the question to reflect my misuse of the terminology.
Dec
19
asked Understanding the random variable definition of Markov chains
Nov
17
awarded  Good Question
Sep
17
comment Is there more than one way to divide the “L”-shaped tromino into four congruent, connected pieces?
I don't suppose you have an idea of how to do this if boundaries are excluded or an opinion on whether a solution besides the one I originally came up with is possible?
Sep
17
comment Is there more than one way to divide the “L”-shaped tromino into four congruent, connected pieces?
+1 This is clever. If it also answered the original polygons version, then I'd award your dad the bounty. Maybe I should anyway...I don't anticipate anyone proving it can't be done in another way with four polygons or coming up with another solution.
Sep
14
comment Is there more than one way to divide the “L”-shaped tromino into four congruent, connected pieces?
@MikePierce Oh gotcha. Yeah I would consider Jon Mark Perry's answer cheating. I'd be interested in a solution using something like the bowtie, but I don't think such a solution would be in line with the spirit of the original question.
Sep
14
comment Is there more than one way to divide the “L”-shaped tromino into four congruent, connected pieces?
@Rick To be fair to Mike, I did mention "parts" in my addendum.
Sep
14
revised Is there more than one way to divide the “L”-shaped tromino into four congruent, connected pieces?
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Sep
14
comment Is there more than one way to divide the “L”-shaped tromino into four congruent, connected pieces?
@MikePierce I'll make the appropriate edit. Thanks.