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Chapter 3: Problem 105

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 \(1 \mathrm{mile}\). 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. Use the Poisson pmf to find \( P(X \leq 16) \) for \( \lambda = 20 \). b. Expected trees: 6,800,000. c. PMF is \( P(X = k) = \frac{e^{-160 \pi} (160 \pi)^k}{k!} \).

Step by step solution

01

Understanding the Poisson Process

The trees are distributed according to a Poisson process with parameter \( \alpha = 80 \) trees per acre. This means the number of trees in any area \( A \) is given by a Poisson distribution with parameter \( \lambda = \alpha \times A \).
02

Calculate Probability for Quarter-Acre Plot

A quarter-acre plot has an area of \( 0.25 \) acres. Thus, the mean number of trees, \( \lambda \), is \( 80 \times 0.25 = 20 \). The probability of finding at most 16 trees in this plot is:\[ P(X \leq 16) = \sum_{k=0}^{16} \frac{e^{-20} \cdot 20^k}{k!} \] This requires calculating the sum of probabilities for \( k = 0 \) to \( 16 \) using the Poisson probability mass function.
03

Compute Expected Number of Trees in the Forest

The entire forest covers 85,000 acres. The expected number of trees in this area can be calculated by multiplying the average number of trees per acre by the total number of acres: \[ E = 80 \times 85,000 = 6,800,000 \] Thus, the expected number of trees in the forest is 6,800,000.
04

Determine PMF of X in Circular Region

The area of a circular region with radius 1 mile is \( \pi \times (1)^2 = \pi \) square miles. Since 1 square mile equals 640 acres, the area is \( 640\pi \) acres. Thus, the mean number of trees, \( \lambda \), in this region is:\[ \lambda = 80 \times 640\pi \approx 160 \pi \]Therefore, \( X \) follows a Poisson distribution with parameter \( \lambda = 160\pi \), and the pmf is \[ P(X = k) = \frac{e^{-160 \pi} \cdot (160 \pi)^k}{k!} \]

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Key Concepts

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

Poisson Distribution
The Poisson distribution is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed interval of time or space. It is used when these events occur independently, and the average number of events ( \( \lambda \) ) is known.
  • It is defined by a single parameter, \( \lambda \), which is the expected number of occurrences that can happen within the given interval.
  • This distribution is perfect for modeling situations like the number of emails received in an hour or, as in this problem, the number of trees in an acre.
In the given exercise, trees in a forest are distributed according to a two-dimensional Poisson process. Here, \( \alpha = 80 \) trees per acre is the given parameter, which denotes the average number of trees per unit area. This makes it feasible to predict tree distribution patterns in varying sizes of plots within the forest.
Probability Mass Function
The Probability Mass Function (PMF) is a formula that gives the probability that a discrete random variable is exactly equal to a particular value. For a Poisson random variable with a parameter \( \lambda \), the PMF is given by:\[P(X = k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!} \]Here:
  • \( e \) is the base of the natural logarithm, approximately equal to 2.718.
  • \( k \) is the actual number of occurrences we want to find the probability for.
  • \( k! \) is the factorial of \( k \), a necessary part of the computation.
In our problem, to find the likelihood of having at most 16 trees on a quarter-acre plot, the PMF is calculated for all values between 0 and 16 and then summed up. This reflects the cumulative probability of having no more than 16 trees in that specific area.
Expected Value
The expected value, often denoted as \( E(X) \) , of a random variable provides us an idea of its long-term average. For a Poisson distribution, the expected value is simply the parameter \( \lambda \) itself. This implies that in any interval defined for a Poisson distribution, \( \lambda \) events are expected to happen on average.
  • For a full forest of 85,000 acres, being given 80 trees per acre, the expected total becomes: \[ E = \alpha \times \, \text{Total Acres} = 80 \times 85,000 = 6,800,000 \] trees.
This means, in general, you should anticipate approximately 6,800,000 trees based on the average density provided by the problem, offering a statistical expectation over the large extent of the forest.
Mathematical Statistics
Mathematical statistics involves the study of statistics using mathematical theories and methods. It includes processing data, making inferences, and developing new statistical techniques. In this context, we employ mathematical statistics to harness the Poisson process in predicting the number of trees in specified areas within a forest.
  • The use of the Poisson distribution is crucial to model occurrences like tree distribution, given their natural unpredictability yet statistically predictable results over many trials.
  • Reliable modeling through mathematical statistics provides insights into scenarios often difficult to measure directly because it allows calculating probabilities for complex geographical and event processes.
Such advanced modeling illustrated here with the Poisson PMF or expected value calculations underpins real-world decisions in forest management and other ecological studies, enhancing our understanding of issues at scale.

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Most popular questions from this chapter

Suppose that you read through this year's issues of the New York Times and record each number that appears in a news article-the income of a CEO, the number of cases of wine produced by a winery, the total charitable contribution of a politician during the previous tax year, the age of a celebrity, and so on. Now focus on the leading digit of each number, which could be \(1,2, \ldots, 8\), or 9 . Your first thought might be that the leading digit \(X\) of a randomly selected number would be equally likely to be one of the nine possibilities (a discrete uniform distribution). However, much empirical evidence as well as some theoretical arguments suggest an alternative probability distribution called Benford's law: $$ \begin{aligned} p(x) &=P(1 \text { st digit is } x)=\log _{10}\left(\frac{x+1}{x}\right), \\ x &=1,2, \ldots, 9 \end{aligned} $$ a. Without computing individual probabilities from this formula, show that it specifies a legitimate pmf. b. Now compute the individual probabilities and compare to the corresponding discrete uniform distribution. c. Obtain the cdf of \(X\). d. Using the odf, what is the probability that the leading digit is at most 3 ? At least 5 ? [Note: Benford's law is the basis for some auditing procedures used to detect fraud in financial reporting-for example, by the Internal Revenue Service.]

For a particular insurance policy the number of claims by a policy holder in 5 years is Poisson distributed. If the filing of one claim is four times as likely as the filing of two claims, find the expected number of claims.

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 amount of royalties earned from the sale of a first edition of 10,000 textbooks 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 coin tosses required for three individuals to obtain a match (HHH or TTT)

. Suppose that \(20 \%\) of all individuals have an adverse reaction to a particular drug. A medical researcher will administer the drug to one individual after another until the first adverse reaction occurs. Define an appropriate random variable and use its distribution to answer the following questions. a. What is the probability that when the experiment terminates, four individuals have not had adverse reactions? b. What is the probability that the drug is administered to exactly five individuals? c. What is the probability that at most four individuals do not have an adverse reaction? d. How many individuals would you expect to not have an adverse reaction, and to how many individuals would you expect the drug to be given? e. What is the probability that the number of individuals given the drug is within 1 standard deviation of what you expect?

Show that \(E(X)=n p\) when \(X\) is a binomial random variable. [Hint: First express \(E(X)\) as a sum with lower limit \(x=1\). Then factor out \(n p\), let \(y=x-1\) so that the remaining sum is from \(y=0\) to \(y=n-1\), and show that it equals 1.]
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