You are a biologist which is interested in modeling the dispersion of sharks wit
ID: 3292126 • Letter: Y
Question
You are a biologist which is interested in modeling the dispersion of sharks within the ocean. You monitor a portion of the ocean that is square and of size 10 km by 10 km. You assume the location of sharks within the ocean can be represented by a 2-dimensional Poisson process with rate 0.05 sharks by km^2. Thus, you don't know in advance the number of sharks that are in the monitored region. Perform a Monte-Carlo simulation with N = 1000 repetitions to estimate the probability that there exists at least two sharks in the lake whose distance between each other is less than 1 km.Explanation / Answer
a) Since the area of the square is 100km2 and the rate of the Poisson process is 0.05/km2 the number of sharks in the square area follows a Poisson distribution with mean = 5. We use the fact that given the number of observations the spatial distribution followd a Uniform distribution on the disc. Therefore the steps in the simulation is
1) Generate a Poisson distribution with mean = 5. Let the generated value be N
2) Generate N i.i.d. Uniform random variables (xi,yi) in the 10 X 10 disc of area 100 1 <=i <= N.
3) Check the number of pairs of sharks which are a distance between 1 km.
4) Repeat the procedure 1000 times. Count the numbers in each case and find the proportion.
After doing a simulation in R the probability was estimated to be equal to 0.397.
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