Throughout our IA, we managed to analyze our statistical modeling with new fields of mathematics. We defined and applied linear algebra with Matrix multiplication, and derived some significant probabilities of the model. We started with 2 by 2 matrix to gain some insight, where it led to the essential property of regular matrix and its role with stationary matrix. For regular Markov Chain, it was crucial to determine the transition matrix, while the initial state did not play much role. Furthermore, we managed to show how the regular matrix could be verified and used to finding the stationary matrix. With this technique, we no longer had to do the tedious multiplication of matrix but find the stationary matrix with inverse matrix multiplication.
We then showed regardless of size of the matrix, although we have to preserve the n by n form for the transition matrix, the concept of regular Markov Chain with stationary matrix worked as well. Hence, we extended the idea of regular Markov chain to our next example with Ryu, then derived the stationary matrix to analyze his performance.
There are thousands of applications of Markov Chain in our current world. The first discovery of Markov chain was from linguistics analysis; back in 1913 Andrew Markov carried a study of how often a vowel is followed by another vowel or a consonant by another consonant in Russian text.
Economics also takes various examples of Markov Chain. Housing pattern of urban area with populations also could be described by analyzing its transition matrix as well as stationary matrix. We can also get the result from the 10-year period to find the probability that someone living in a single-family dwelling is doing so for 10-year period. Such analysis is important for urban development.
It extends to even Politics; you can analyze the voting trends from the initial state of a particular period and extend it to general trends for longer period. Then we can analyze accordingly for the percent of each type of voter at the end of each month of the year, which gives a great tool for political party to prepare for their elections.
Markov Chain gave a much insight on how mathematics could be applied into the real-life problem. For the contents in pure mathematics aspect, it was hard to formalize the use of the mathematics. However, fields such as statistics and probability I related well personally to the field with real life example. This method of learning was in fact more challenging yet fruitful. Having the understanding of a study and immediately applying it to the real-life problem not only expanded my horizon of understanding with mathematics, but also made me more active in gaining the knowledge. Learning statistics could definitely encourage students’ participation in learning mathematics as it is more tangible than conceptualizing abstract theorems. By learning Markov chain, I not only enjoyed the calculation, but also the application; the most important application of all; how to think. “Mathematics is not just solving for x. It’s also figuring out why.”
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