Question

How many people do you expected to arrive during a 45min period?Suppose that trees are distributed in a forest according to a two-dimensional Poisson process with parameter λ, the expected number of trees per acree, equal to 80.

1. What is the probability that in a certain quarter-acre-plot, there will be at most 16 trees?
2. If the forest covers 85,000 acres, what is the expected number of trees in the forest?
3. Suppose you selected a point in the forest and construct a circle of radius 0.1 mile. Let X = the number of trees within that circular region. What is the pmf of X? (Hint: 1 sq. mile = 640 acres.)

Answer 1

a) The average number of trees per acre is given here as 80. Therefore the number of acres per quarter acre is computed here as:
= 80 /4 = 20 trees per quarter acre.

The probability that there will be at most 16 tress in a certain quarter acre plot is computed here as:
P( X <= 16)

This is computed in EXCEL here as:
=poisson.dist(16,20,TRUE)

0.2211 is the output here. Therefore 0.2211 is the required probability here.

b) The expected number of trees in a 85,000 acres forest cover is computed here as:

= 85,000*Average number of trees per acre

= 85,000*80

= 6800000

therefore 6800000 trees are expected to be in 85,000 acres of forest cover.

c) The area of the circle here is computed as:

Average number of trees in A is computed here as:
= 20.1062*80 = 1608.4954

The PMF for the number of trees is therefore given here as:

This is the required PMF for X here.