Stochastic Modelling Question. Please do all Batman chases the Joker around the

Question

Stochastic Modelling Question. Please do all

Batman chases the Joker around the vertices of a square. At each time step. Batman steps clockwise with probability p elementof (0, 1), and anticlockwise with probability 1 - p, while the Joker stays where he is with probability q elementof [0, 1], and steps clockwise with probability (1 - q)s and anticlockwise with probability (1 - q)(1 - s), where s elementof (0, 1). Batman catches the Joker if they reach the same vertex at the same time. All steps are taken independently of previous steps. Starting from opposite corners of the square: (a) what is the expected time until Batman catches the Joker when q = 1? (b) if q = 0, find the expected time until Batman catches the Joker, and for fixed p, find the maximum possible value for this quantity (i.e. optimize over s). (c) When p = 1/2, find the expected time until Batman catches the Joker.

Explanation / Answer

a) q =1

means Joker stays where he is that's fixed corner

possible way to catch are

- in x= 2 steps i.e. both steps in clockwise or both anticlockwise [ p*p and (1-p)*(1-p) ]

- in x= 3 steps not possible

- in x=4 steps i.e. 3 clockwise and 1 anticlockwise or 1 clockwise and 3 anticlockwise

E(x)= 2(p*p + (1-p)*(1-p)) + 4(p3(1-p) + p(1-p)3)

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