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Question

# Suppose that trees are distributed in a forest according to a two-dimensional Poisson process with parameter $\alpha$, the expected number of trees per acre, equal to 80. a. What is the probability that in a certain quarter-acre plot, there will be at most 16 trees? b. If the forest covers 85,000 acres, what is the expected number of trees in the forest? c. Suppose you select a point in the forest and construct a circle of radius .1 mile. Let X=the number of trees within that circular region. What is the pmf of X? [Hint: 1 sq mile=640 acres.]

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$\text{\textcolor{#4257b2}{\textbf{Proposition:}}}$ Number of events during a time interval of length t can be modeled using Poisson Random Variable with parameter $\mu = \alpha t$. This indicates that

$P_k(t) = e^{-\alpha t} \cdot \frac{(\alpha t)^k}{k!}.$

$\text{\textcolor{#4257b2}{\textbf{Poisson process}}}$

We are given that

$\alpha = 80.$

$\textcolor{#19804f}{\textbf{(a):}}$

In this case, since a quarter is 0.25 of a acre, the mean parameter would be

$\alpha \cdot 0.25 = 80 \cdot 0.25 = 20.$

Using this, the following is true

$\begin{equation*} \begin{split} P_{16}(0.25) = \textcolor{#4257b2}{P(X \leq 16)} = F(16;20) \overset{\textcolor{#c34632}{(1)}}{=} \textcolor{#4257b2}{0.221}; \end{split} \end{equation*}$

$\textcolor{#c34632}{(1):}$ Appendix Table A.2 contains the Poisson cdf $F(x;\mu)$.

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