The height of a midge above a lawn fluctuates in such a way that the random variable. H(t) height at time t height at time 0 may be modelled as geometric Brownian motion, derived from ordinary Brownian motion {X(t); t≥ 0} with diffusion coefficient o² = 0.27 per minute through the relationship H(t) = exp X(t). (a) The midge is 25 centimetres above the lawn at time 0. Calculate the probability that after three minutes the midge will be more than 40 centimetres above the lawn. (b) Initially, the midge is 25 centimetres above the lawn. Use the numbers 0.7879, -0.9573, 1.4018, which are random observations from the standard normal distribution N(0, 1), to simulate the height of the midge above the lawn at 20-second intervals for one minute. Give the heights above the lawn to the nearest centimetre.
The height of a midge above a lawn fluctuates in such a way that the random variable. H(t) height at time t height at time 0 may be modelled as geometric Brownian motion, derived from ordinary Brownian motion {X(t); t≥ 0} with diffusion coefficient o² = 0.27 per minute through the relationship H(t) = exp X(t). (a) The midge is 25 centimetres above the lawn at time 0. Calculate the probability that after three minutes the midge will be more than 40 centimetres above the lawn. (b) Initially, the midge is 25 centimetres above the lawn. Use the numbers 0.7879, -0.9573, 1.4018, which are random observations from the standard normal distribution N(0, 1), to simulate the height of the midge above the lawn at 20-second intervals for one minute. Give the heights above the lawn to the nearest centimetre.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Transcribed Image Text:The height of a midge above a lawn fluctuates in such a way that the
random variable.
H(t)
height at time t
height at time 0
may be modelled as geometric Brownian motion, derived from ordinary
Brownian motion {X(t); t≥ 0} with diffusion coefficient o² = 0.27 per
minute through the relationship H(t) = exp X(t).
(a) The midge is 25 centimetres above the lawn at time 0. Calculate the
probability that after three minutes the midge will be more than
40 centimetres above the lawn.
(b) Initially, the midge is 25 centimetres above the lawn. Use the numbers
0.7879, -0.9573, 1.4018, which are random observations from the
standard normal distribution N(0, 1), to simulate the height of the
midge above the lawn at 20-second intervals for one minute. Give the
heights above the lawn to the nearest centimetre.
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