May 29th, 2015
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APPLICATION TO MATRIX ALGEBRA1) MARKOV CHAINSSuppose there is a physical or mathematical system that has n possible states and at any one time, the system is in one and only one of its n states. The simplest example is that of independent and identically distributed random variables. In this case, Xn describes the outcome of a repeated experiment, at the nth trial. In such systems, the future states of the systems is independent of past and present states. In most other systems that arise in practice past and present states inuence the evolution of the system, hence the future states. Obviously, if present and past systems determine the future states of the system uniquely, it means that there is not randomness in the system from that time on. Such a system is called Markov Chain or Markov Process. A Markov Chains is a process that consists of a finite number of states and some known possibilities Pij where Pij is the probability of moving from state j to state i. Let us clarify the definition of this process using the example below:Example :Age Distribution of Trees in a ForestTrees in a forest are assumed in this simple model to fall into four age groups: b(k) denotes the number of baby trees in the forest (age group 0 15 years) at a given time period k, similarly y(k), m(k) and o(k) denote the number of young trees (16-30 years of age), middle-aged trees (age 31-45) and old trees (older than 45 years of age) respectively. The length of on

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