This question is intended to assess your understanding of ordinary Brownian motion. Each day, two children leave their h

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This Question Is Intended To Assess Your Understanding Of Ordinary Brownian Motion Each Day Two Children Leave Their H 1 (220.6 KiB) Viewed 37 times

This question is intended to assess your understanding of ordinary Brownian motion. Each day, two children leave their home at 9 am to play along the bank of a stream that runs past their home. Their distance (in metres) upstream from home after t hours is denoted by X(t), and may be reasonably modelled as ordinary Brownian motion {X(t);t ≥ 0} with diffusion coefficient ² = 450 m² per hour. (a) Calculate the probability that two hours after leaving home, the children will be within 50 metres of their home. (b) One day, the children are observed 30 metres upstream from their home at 11 am. Calculate the probability that at midday the children will be upstream from their home. (c) On another day, the children are observed to return home at midday. Calculate the probability that at 11 am the children were within 20 metres of their home. (d) There is a large willow 60 metres upstream from the children's home. Calculate the probability that the children will first reach the willow more than two hours after leaving home.