One Dimensional Random Walk

Think of it this way. We now confine our particle to wander back and forth along a linear path. A bell-shaped Gaussian distribution defines the probability of the particle being at some specified distance from the starting point after a certain number of steps. The more steps taken, the wider the curve. Indeed our expected average distance from the start is just the length of each step times the square root of the number of steps taken.

A particle being pounded on incessantly goes with the flow, so imagine a particle between two organs. Every minute with a fifty percent probability, P = 1/2, it jiggles either ten units towards 'The Thorax' or with probability of a third, P = 1/3, it drifts towards 'The Midgut' or it remains for the interval where it is with P = 1/6. Our particle takes a random walk in one dimension, and its movements with time form a finite Markov chain. Assume also the midgut and thorax hold the particle if it arrives there. Knowing the distance between the midgut and thorax and our particle's starting position, we ask what place is it likely to reach first, and how long will it take to get there.

If the midgut and thorax are fifty units apart, and our particle is originally twenty units from the thorax, we can label its potential stops as E1 to E6 with our two extremes being the midgut and thorax. We give E4 as the vector x = (0, 0, 0, 1, 0, 0) in which the ith component of this vector is the probability that the particle is initially at Ei. Vectors (0, 0, 1/2, 1/6, 1/3, 0) and (0, 1/4, 1/6, 13/36, 1/9, 1/9) are our particle's positions after one minute and then two minutes elapse. Using a transition matrix, we can imagine its location after k minutes:

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