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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 added 143 characters in body 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 deleted 8 characters in body Dec 19 revised Understanding the random variable definition of Markov chains added 13 characters in body 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? deleted 15 characters in body; edited title 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.