Compute the quantity requested in each problem. Give an exact answer if possible

Question

Compute the quantity requested in each problem. Give an exact answer if possible; otherwise, round to three places after the decimal. If you use a Geometric or Binomial distribution, say so and identify the parameter(s). If you use a CDF, mention how you computed it (chart or command).

4. Suppose a virtual landscape is procedurally generated so that each 100 by 100 meter square has an average of seven “points of interest” to an explorer (e.g., treasure, readable, or cave entrance) that appear according to a Poisson distribution. If an optimal number of “points of interest” is at least 3 but at most 10, what percentage of 100 by 100 meter squares will have an optimal number of such points?

Explanation / Answer

To answer question numbered 4 I have used a CDF function from R. The name of the function is ppois(q, lambda) which returns Prob(X <= q) where X ~ Poisson(lambda).

To answer the question about what percentage of the 100 by 100 meter squares will have optimal number of points of interest we need to find out the probability that X is in between 3 and 10 where X follows a Poisson(7) distribution as the avergae number of points is 7 for a 100 by 100 sq. meter area.

Prob(3 <= X <= 10) = Prob(X <= 10) - Prob(X <= 2) = ppois(10,lambda=7) - ppois(2, lambda=7) = 0.871843.

Hence nearly 87.18% of 100 by 100 sq. m. ground have optimal number of points of interest.

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