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Chapter 3: Problem 91

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.]

Short Answer

Expert verified
a. Calculate the cumulative Poisson probability for \(k \leq 16\); b. Expected number of trees is 6,800,000; c. PMF is Poisson with \(\lambda \approx 1608.48\).

Step by step solution

01

Understanding the Poisson Process

In a two-dimensional Poisson process, the number of events (trees in this case) in an area follows a Poisson distribution. The parameter \( \lambda \) of the Poisson distribution is the expected number of events in the area of interest. Given \( \alpha = 80 \) trees per acre, this parameter will be scaled according to the size of the plot.
02

Calculate \( \lambda \) for a quarter-acre plot

For part (a), we have a quarter-acre plot. Calculate \( \lambda \) by multiplying \( \alpha \) by the size of the plot: \[ \lambda = 80 \times \frac{1}{4} = 20 \]Thus, the number of trees in the quarter-acre plot follows a Poisson distribution with \( \lambda = 20 \).
03

Calculate the Probability for at Most 16 Trees

To find the probability of having at most 16 trees, use the cumulative Poisson probability:\[ P(X \leq 16) = \sum_{k=0}^{16} \frac{e^{-20} \cdot 20^k}{k!} \]This involves summing up the individual Poisson probabilities from 0 to 16 trees.
04

Expected Number of Trees in the Entire Forest

For part (b), multiply the area of the forest (85,000 acres) by the average number of trees per acre to get the expected value:\[ \text{Expected Trees} = 80 \times 85,000 = 6,800,000 \]
05

Finding \( \lambda \) for the Circular Region

For part (c), convert the circular area into acres. The radius is 0.1 mile, so the area in square miles is \( \pi \times (0.1)^2 \) square miles. Convert this to acres using the relation 1 square mile = 640 acres:\[ \text{Area} = \pi \times (0.1)^2 \times 640 \approx 20.106 \text{ acres} \]Calculate \( \lambda \) for this area:\[ \lambda = 80 \times 20.106 \approx 1608.48 \]
06

PMF of \(X\)

The number of trees \( X \) within the circular region follows a Poisson distribution with parameter \( \lambda \approx 1608.48 \). Thus, the probability mass function (pmf) of \( X \) is:\[ P(X = k) = \frac{e^{-1608.48} \cdot 1608.48^k}{k!} \text{ for } k = 0, 1, 2, \ldots \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Distribution
In the context of the Poisson process, understanding the probability distribution is crucial. A Poisson distribution is used to model the number of events occurring within a fixed interval of time or space. It is defined by a single parameter \( \lambda \), which represents the expected number of events. When you're dealing with spatial distribution, like trees in a forest, this parameter becomes essential. It tells us, on average, how many trees you might expect to find in a given plot.
  • The Poisson distribution is discrete, meaning it calculates the probability of a specified number of discrete events.
  • It is relevant for counts of events in space or time, such as trees per acre.
  • Key characteristic: the occurrences are independent, which implies that the number of trees in one part of the forest doesn't influence those in another part.
Understanding these foundational elements will aid in solving more complex problems involving Poisson processes.
Parameter Calculation
Calculating the parameter \( \lambda \) is a pivotal step when dealing with a Poisson process. In problems involving spatial distribution, \( \lambda \) is calculated by multiplying the average number of occurrences (like trees per acre) by the size of the area in interest. For example, if the average number of trees per acre, \( \alpha \), is 80, then for a quarter-acre plot, you calculate \( \lambda \) as follows:
\[ \lambda = 80 \times \frac{1}{4} = 20 \]
  • The parameter \( \lambda \) is adjustable based on the area's size.
  • This makes the model flexible for various spatial queries, as seen in part (a) of the original problem.
  • Understanding \( \lambda \) allows us to predict the distribution of trees across different plot sizes.
Parameter calculation ensures you have the right \( \lambda \), appropriate for solving the probability equations of the Poisson process.
Two-Dimensional Poisson
A two-dimensional Poisson process expands the concept of event occurrences into spatial regions, rather than just time intervals. This means events (like trees) are spread across a plane (the forest), and we are interested in how the density of these events changes across different areas.

  • Two-dimensional Poisson processes are crucial for problems involving areas instead of simple linear distances.
  • The calculation of \( \lambda \) in this context depends on the density of the events per unit area and the total area.
  • They allow modeling of various real-world spatial distribution problems, such as the number of trees in a forest or the distribution of stars in a galaxy.
In our exercise, the two-dimensional aspect is represented by different units of measure (acres and square miles) for the forest and changes in plot size.
Expected Value Calculation
The expected value in a Poisson process gives a sense of the average number of occurrences you can expect in a given space. It's calculated as \( \lambda \times \text{Area} \), where \( \lambda \) is the average number of events per unit area.

In part (b) of our example:
  • The entire forest covers 85,000 acres with an average of 80 trees per acre.
  • The expected number of trees is calculated by multiplying these two: \[ \text{Expected Trees} = 80 \times 85,000 = 6,800,000 \]
  • Understanding expected value helps in predicting overall trends across larger datasets or areas.
For both planners and ecologists, calculating the expected value can assist in resource management and environmental analysis.

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Most popular questions from this chapter

Twenty percent of all telephones of a certain type are submitted for service while under warranty. Of these, \(60 \%\) can be repaired, whereas the other \(40 \%\) must be replaced with new units. If a company purchases ten of these telephones, what is the probability that exactly two will end up being replaced under warranty?

Each time a component is tested, the trial is a success ( \(S\) ) or failure \((F)\). Suppose the component is tested repeatedly until a success occurs on three consecutive trials. Let \(Y\) denote the number of trials necessary to achieve this. List all outcomes corresponding to the five smallest possible values of \(Y\), and state which \(Y\) value is associated with each one.

For each random variable defined here, describe the set of possible values for the variable, and state whether the variable is discrete. a. \(X=\) the number of unbroken eggs in a randomly chosen standard egg carton b. \(Y=\) the number of students on a class list for a particular course who are absent on the first day of classes c. \(U=\) the number of times a duffer has to swing at a golf ball before hitting it d. \(X=\) the length of a randomly selected rattlesnake e. \(Z=\) the sales tax percentage for a randomly selected amazon.com purchase f. \(Y=\) the \(\mathrm{pH}\) of a randomly chosen soil sample g. \(X=\) the tension (psi) at which a randomly selected tennis racket has been strung h. \(X=\) the total number of times three tennis players must spin their rackets to obtain something other than \(U U U\) or \(D D D\) (to determine which two play next)

A contractor is required by a county planning department to submit one, two, three, four, or five forms (depending on the nature of the project) in applying for a building permit. Let \(Y=\) the number of forms required of the next applicant. The probability that \(y\) forms are required is known to be proportional to \(y\)-that is, \(p(y)=k y\) for \(y=1, \ldots, 5\). a. What is the value of \(k\) ? [Hint: \(\left.\Sigma_{y=1}^{5} p(y)=1\right]\) b. What is the probability that at most three forms are required? c. What is the probability that between two and four forms (inclusive) are required? d. Could \(p(y)=y^{2} / 50\) for \(y=1, \ldots, 5\) be the pmf of \(Y\) ?

A certain brand of upright freezer is available in three different rated capacities: \(16 \mathrm{ft}^{3}, 18 \mathrm{ft}^{3}\), and \(20 \mathrm{ft}^{3}\). Let \(X=\) the rated capacity of a freezer of this brand sold at a certain store. Suppose that \(X\) has pmf \begin{tabular}{l|ccc} \(x\) & 16 & 18 & 20 \\ \hline\(p(x)\) & \(.2\) & \(.5\) & \(.3\) \end{tabular} a. Compute \(E(X), E\left(X^{2}\right)\), and \(V(X)\). b. If the price of a freezer having capacity \(X\) is \(70 X-650\), what is the expected price paid by the next customer to buy a freezer? c. What is the variance of the price paid by the next customer? d. Suppose that although the rated capacity of a freezer is \(X\), the actual capacity is \(h(X)=X-.008 X^{2}\). What is the expected actual capacity of the freezer purchased by the next customer?
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