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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

It is known that\({\rm{80\% }}\)of all brand A extremal hard drives work in a satisfactory manner throughout the warranty period (are "successes"). Suppose that\({\rm{n = 15}}\)drives are randomly selected. Let\({\rm{X = }}\)the number of successes in the sample. The statistic\({\rm{X/n}}\)is the sample proportion (fraction) of successes. Obtain the sampling distribution of this statistic. (Hint: One possible value of\({\rm{X/n is }}{\rm{.2}}\), corresponding to\({\rm{X = 3}}\). What is the probability of this value (what kind of\({\rm{rv}}\)is\({\rm{X}}\))?)

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.

Let \({\rm{X}}\) be the number of packages being mailed by a randomly selected customer at a certain shipping facility. Suppose the distribution of \({\rm{X}}\) is as follows:

a. Consider a random sample of size \({\rm{n = 2}}\) (two customers), and let \({\rm{\bar X}}\) be the sample mean number of packages shipped. Obtain the probability distribution of\({\rm{\bar X}}\).

b. Refer to part (a) and calculate\({\rm{P(\bar X\pounds2}}{\rm{.5)}}\).

c. Again consider a random sample of size\({\rm{n = 2}}\), but now focus on the statistic \({\rm{R = }}\) the sample range (difference between the largest and smallest values in the sample). Obtain the distribution of\({\rm{R}}\). (Hint: Calculate the value of \({\rm{R}}\) for each outcome and use the probabilities from part (a).)

d. If a random sample of size \({\rm{n = 4}}\) is selected, what is \({\rm{P(\bar X\pounds1}}{\rm{.5)}}\) ? (Hint: You should not have to list all possible outcomes, only those for which\({\rm{\bar x\pounds1}}{\rm{.5}}\).)

The mean weight of luggage checked by a randomly selected tourist-class passenger flying between two cities on a certain airline is\({\bf{40}}\)lb, and the standard deviation is\({\bf{10}}\)lb. The mean and standard deviation for a business class passenger is\({\bf{30}}\)lb and\({\bf{6}}\)lb, respectively.

a. If there are\({\bf{12}}\)business-class passengers and\({\bf{50}}\)tourist-class passengers on a particular flight, what is the expected value of total luggage weight and the standard deviation of total luggage weight?

b. If individual luggage weights are independent, normally distributed RVs, what is the probability that total luggage weight is at most\({\bf{2500}}\)lb?

a. Compute the covariance between \({\rm{X}}\) and\({\rm{Y}}\).

b. Compute the correlation coefficient \({\rm{\rho }}\) for this \({\rm{X}}\) and \({\rm{Y}}\).
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