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Chapter 3: Q110E (page 138)

Grasshoppers are distributed at random in a large field according to a Poisson process with parameter a \({\rm{\alpha = 2}}\) per square yard. How large should the radius \({\rm{R}}\) of a circular sampling region be taken so that the probability of finding at least one in the region equals \({\rm{.99}}\)?

Short Answer

Expert verified

The radius \({\rm{R}}\) of a circular sampling region should be taken \({\rm{R = 0}}{\rm{.85612}}\) so that the probability of finding at least one in the region equals \({\rm{.99}}\).

Step by step solution

01

Concept Introduction

Probability refers to the likelihood of a random event's outcome. This word refers to determining the likelihood of a given occurrence occurring.

Proposition: Number of events during a time interval of length\({\rm{t}}\)can be modelled using Poisson Random Variable with parameter\({\rm{\mu = \alpha t}}\). This indicates that 鈥�

\({{\rm{P}}_{\rm{k}}}{\rm{(t) = }}{{\rm{e}}^{{\rm{ - \alpha t}}}}\frac{{{{{\rm{(\alpha t)}}}^{\rm{k}}}}}{{{\rm{x!}}}}\)

It is also known as Poisson process (not formally defined).

02

Radius of the region

It is given that 鈥�

\({\rm{t = }}\pi {{\rm{R}}^{\rm{2}}}\)(circular sampling region).

This means that 鈥�

\({\rm{\mu = \alpha }} \cdot {\rm{t = 2}} \cdot \pi {{\rm{R}}^{\rm{2}}}\)

Therefore, it is obtained 鈥�

\(\begin{array}{c}{\rm{P(X}} \ge {\rm{1) = 1 - P(X = 0)}}\\{\rm{ = 1 - }}{{\rm{P}}_{\rm{0}}}\left( {\pi {{\rm{R}}^{\rm{2}}}} \right)\\{\rm{ = 1 - }}{{\rm{e}}^{{\rm{ - 2}} \cdot \pi {{\rm{R}}^{\rm{2}}}}} \cdot \frac{{{{\left( {{\rm{2}} \cdot \pi {{\rm{R}}^{\rm{2}}}} \right)}^{\rm{0}}}}}{{{\rm{0!}}}}\\{\rm{ = 1 - }}{{\rm{e}}^{{\rm{ - 2}} \cdot \pi {{\rm{R}}^{\rm{2}}}}}\end{array}\)

From the relation 鈥�

\({\rm{P(X}} \ge {\rm{1) = 0}}{\rm{.99}}\)

It is obtained that 鈥�

\(\begin{array}{c}{\rm{1 - }}{{\rm{e}}^{{\rm{ - 2}} \cdot \pi {{\rm{R}}^{\rm{2}}}}}{\rm{ = 0}}{\rm{.99}}\\{{\rm{e}}^{{\rm{ - 2}} \cdot \pi {{\rm{R}}^{\rm{2}}}}}{\rm{ = 0}}{\rm{.01}}\\{\rm{ - 2}} \cdot \pi {{\rm{R}}^{\rm{2}}}{\rm{ = ln0}}{\rm{.01}}\\{{\rm{R}}^{\rm{2}}}{\rm{ = - }}\frac{{{\rm{ln0}}{\rm{.01}}}}{{{\rm{2\pi }}}}\\{\rm{R = 0}}{\rm{.85612}}\end{array}\)

Therefore, the value is obtained as \({\rm{R = 0}}{\rm{.85612}}\).

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Most popular questions from this chapter

A manufacturer of integrated circuit chips wishes to control the quality of its product by rejecting any batch in which the proportion of defective chips is too high. To this end, out of each batch (10,000 chips), 25 will be selected and tested. If at least 5 of these 25 are defective, the entire batch will be rejected.

a. What is the probability that a batch will be rejected if 5% of the chips in the batch are in fact defective?

b. Answer the question posed in (a) if the percentage of defective chips in the batch is \({\bf{10}}\% \).

c. Answer the question posed in (a) if the percentage of defective chips in the batch is \({\bf{20}}\% \).

d. What happens to the probabilities in (a)鈥�(c) if the critical rejection number is increased from 5 to \({\bf{6}}\)?

An individual named Claudius is located at the point 0 in the accompanying diagram. Using an appropriate randomization device (such as a
tetrahedral die, one having four sides), Claudius first moves to one of the four locations B1, B2, B3, B4. Once at one of these locations, another randomization device is used to decide whether Claudius next returns to 0 or next visits one of the other two adjacent points. This process then continues; after each move, another move to one of the (new) adjacent points is determined by tossing an appropriate die or coin.

a. Let X = the number of moves that Claudius makes before first returning to 0. What are possible values of X? Is X discrete or continuous?

b. If moves are allowed also along the diagonal paths connecting 0 to A1, A2, A3, and A4, respectively, answer the questions in part (a).

Suppose that trees are distributed in a forest according to a two-dimensional Poisson process with parameter\({\rm{\alpha }}\), the expected number of trees per acre, equal to\({\rm{80}}\). a. What is the probability that in a certain quarter-acre plot, there will be at most\({\rm{16}}\)trees? b. If the forest covers\({\rm{85,000}}\)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\({\rm{.1}}\)mile. Let X = the number of trees within that circular region. What is the pmf of X? (Hint:\({\rm{1}}\)sq mile\({\rm{ = 640}}\)acres.)

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

Of the people passing through an airport metal detector, .5% activate it; let \(X = \) the number among a randomly selected group of 500 who activate the detector.

a. What is the (approximate) pmf of X?

b. Compute \({\bf{P}}(X = 5)\)

c. Compute \({\bf{P}}(5 \le X)\)
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