Question

Suppose that trees are distributed in a forest according to a two-dimensional Poisson process with parameter α, the expected number of trees per acre, equal to 40. (a) What is the probability that in a certain quarter-acre plot, there will be at most 15 trees? (Round your answer to three decimal places.) (b) If the forest covers 70,000 acres, what is the expected number of trees in the forest? trees (c) Suppose you select 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

Answer:

a) 0.951

b) 2,800,000

c) $P(T(R)=n)=\frac{(40*20.106)^ne^{-40*20.106}}{n!}=\frac{(804.248)^ne^{804.248}}{n!}$

Step-by-step explanation:

Let R be a bounded and measurable region in the forest and denote with

|R| = area of R in acres

Let T(R) be a discrete random variable that measures the number of trees in the region R.

If the trees are distributed according to a two-dimensional Poisson process with the expected number of trees per acre equals to 40, then  

$\large P(T(R)=n)=\frac{(40|R|)^ne^{-40|R|}}{n!}$

(a) What is the probability that in a certain quarter-acre plot, there will be at most 15 trees?

In this case R is a region with an area of 1/4 acres,

$\large P(T(R)\leq 15)=\sum_{k=0}^{15}\frac{(40*1/4)^ke^{-40*1/4}}{k!}=e^{-10}\sum_{k=0}^{15}\frac{10^k}{k!}$

We can compute this with a spreadsheet and we get

$\large \boxed{P(T(R)\leq 15)=0.951}$

(b) If the forest covers 70,000 acres, what is the expected number of trees in the forest?

Given that the expected number of trees per acre equals 40, the expected number of trees in 70,000 acres equals  

40*70,000 = 2,800,000 trees.

(c) Suppose you select 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?

Now R is a circle of radius 0.1 mile, so its area equals

$\large |R|=\pi (0.1)^2=0.0314\;squared\;miles$

Since 1 squared mile = 640 acres,

0.0314  squared miles = 0.0314*640 = 20.106 acres, so the pmf of R would be

$\large P(T(R)=n)=\frac{(40*20.106)^ne^{-40*20.106}}{n!}=\frac{(804.248)^ne^{804.248}}{n!}$