Sheldon Ross 10: Example 4.05 (A Random Walk Model)

Question: A Markov chain whose state space is given by the integers i = 0,±1,±2, . . . is said to be a random walk if, for some number 0 < p < 1, Pi,i+1 = p = 1 − Pi,i−1, i = 0,±1, . . .
The preceding Markov chain is called a random walk for we may think of it as being a model for an individual walking on a straight line who at each point of time either takes one step to the right with probability p or one step to the left with probability 1 − p.


Analytical Solution

N/A


Simulation Solution

This is not actually a solution since nothing has been asked to be solved in the example. This would be a preview of what we might see in this chapter. This one is a representation of a random walk and each of the element in the grid shows the simulation when the bias to the right is equal to p. As the value of p (the probability of towards the right) increases, we see the progression of the paths accordingly.  Note that the randomness in the y-direction is intentional and to just represent the path. It has no significance for the left and right bias.

  • p: the probability of moving to the right
  • 1 – p: the probability of moving to the left
  • All random walks start at {0, 0}


Code

Grid[Partition[#, 3], Frame -> All, FrameStyle -> Gray] &[
  Table[With[{p = probability, range = 80},
    Overlay[{Graphics[
      Table[{Opacity@0.5, Thickness@0.0,
        Line[Accumulate[{RandomChoice[{p, 1 - p} -> {-1, 1}],
          RandomChoice[{-1, 1}]} & /@ Range[100]]]}, 1000],
      PlotRange -> ConstantArray[{-range, range}, 2],
      ImageSize -> 262], "p = " <> ToString[p]}]], {probability,
    Range[0.1, 0.9, 0.1]}]]

End of the post


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