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Chapter 5: Q25E (page 219)

A surveyor wishes to lay out a square region with each side having length\({\rm{L}}\). However, because of a measurement error, he instead lays out a rectangle in which the north-south sides both have length \({\rm{X}}\) and the east-west sides both have length\({\rm{Y}}\). Suppose that \({\rm{X}}\) and \({\rm{Y}}\) are independent and that each is uniformly distributed on the interval \({\rm{(L - A,L + A)}}\) (where \({\rm{0 < A < L}}\) ). What is the expected area of the resulting rectangle?

Short Answer

Expert verified

The expected area of the resulting rectangle is \({\rm{E(XY) = }}{{\rm{L}}^{\rm{2}}}{\rm{.}}\)

Step by step solution

01

Definition

Probability simply refers to the likelihood of something occurring. We may talk about the probabilities of particular outcomes鈥攈ow likely they are鈥攚hen we're unclear about the result of an event. Statistics is the study of occurrences guided by probability.

02

Determining the sample size

We are given a square region with a length of a side\({\rm{L}}\). However, a surveyor lands a rectangle. We have two sides with length \({\rm{X}}\) and two with length\({\rm{Y}}\).

Length of north-south sides is somewhere in interval\({\rm{(L - A,L + A)}}\), the same for east-west sides. This means that the length can be any number between \({\rm{L - A}}\) and\({\rm{L + A}}\). Because the given random variables are uniformly distributed on interval\({\rm{(L - A,L + A)}}\), the expected values are the midpoint - which is

\(\frac{{{\rm{L - A + L + A}}}}{{\rm{2}}}{\rm{ = L}}{\rm{.}}\)

An area of a rectangle is measured as \({\rm{a \times b}}\). In our case it is\({\rm{X \times Y}}\). Therefore, we need expectation of random variable \({\rm{X \times Y}}\), which is

\({\rm{E(XY)}}\mathop {\rm{ = }}\limits^{{\rm{(1)}}} {\rm{E(X)E(Y) = L \times L = }}{{\rm{L}}^{\rm{2}}}\)

(1): for two independent random variables \({\rm{E(XY) = E(X)E(Y)}}\).

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Most popular questions from this chapter

The time taken by a randomly selected applicant for a mortgage to fill out a certain form has a normal distribution with mean value \({\rm{10}}\) min and standard deviation \({\rm{2}}\) min. If five individuals fill out a form on one day and six on another, what is the probability that the sample average amount of time taken on each day is at most \({\rm{11}}\) min?

Refer to Exercise, and suppose that the Xi 鈥檚 are independent with each one having a normal distribution. What is the probability that the total volume shipped is at most\(100,000f{t^3}\)?

The difference between the number of customers in line at the express checkout and the number in line at the super-express checkout is\({{\rm{X}}_{\rm{1}}}{\rm{ - }}{{\rm{X}}_{\rm{2}}}\). Calculate the expected difference.

Young鈥檚 modulus is a quantitative measure of stiffness of an elastic material. Suppose that for aluminum alloy sheets of a particular type, its mean value and standard deviation are \({\rm{70 GPa}}\) and \({\rm{1}}{\rm{.6 GPa}}\), respectively (values given in the article 鈥淚nfluence of Material Properties Variability on Springback and Thinning in Sheet Stamping Processes: A Stochastic Analysis鈥� (Intl. J. of Advanced Manuf. Tech., \({\rm{2010:117 - 134}}\))).

  1. If \({\rm{\bar X}}\) is the sample mean young鈥檚 modulus for a random sample of \({\rm{n = 16}}\)sheets, where is the sampling distribution of \({\rm{\bar X}}\)centered, and what is the standard deviation of the \({\rm{\bar X}}\)distribution?
  2. Answer the questions posed in part (a) for a sample size of \({\rm{n = 64}}\)sheets.
  3. For which of the two random samples, the one of part (a) or the one of part (b), is \({\rm{\bar X}}\) more likely to be within \({\rm{1GPa}}\) of \({\rm{70 GPa}}\)? Explain your reasoning.

Question: The number of customers waiting for gift-wrap service at a department store is an rv X with possible values \({\rm{0,1,2,3,4}}\)and corresponding probabilities \({\rm{.1,}}{\rm{.2,}}{\rm{.3,}}{\rm{.25,}}{\rm{.15}}{\rm{.}}\)A randomly selected customer will have \({\rm{1,2}}\),or \({\rm{3}}\) packages for wrapping with probabilities \({\rm{.6,}}{\rm{.3,}}\)and \({\rm{.1,}}\)respectively. Let \({\rm{Y = }}\)the total number of packages to be wrapped for the customers waiting in line (assume that the number of packages submitted by one customer is independent of the number submitted by any other customer).

a. Determine \({\rm{P(X = 3,Y = 3)}}\), i.e., \({\rm{P(3,3)}}\).

b. Determine \({\rm{p(4,11)}}\).
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