91Ó°ÊÓ

The joint probability di\({\rm{Y}} \le {\rm{1) = 0}}{\rm{.12}}\)stribution of the number \({\rm{X}}\) of cars and the number \({\rm{Y}}\) of buses per signal cycle at a proposed left-turn lane is displayed in the accompanying joint probability table.

a. What is the probability that there is exactly one car and exactly one bus during a cycle?

b. What is the probability that there is at most one car and at most one bus during a cycle?

c. What is the probability that there is exactly one car during a cycle? Exactly one bus?

d. Suppose the left-turn lane is to have a capacity of five cars, and that one bus is equivalent to three cars. What is t\({\rm{p(x,y)}} \ge {\rm{0}}\)e probability of an overflow during a cycle?

e. Are \({\rm{X}}\) and \({\rm{Y}}\) independent rv’s? Explain.

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?

Suppose the proportion of rural voters in a certain state who favor a particular gubernatorial candidate is\(.{\bf{45}}\)and the proportion of suburban and urban voters favouring the candidate is\(.{\bf{60}}\). If a sample of\({\bf{200}}\)rural voters and\({\bf{300}}\)urban and suburban voters is obtained, what is the approximate probability that at least\(\;{\bf{250}}\)of these voters favour this candidate?

Garbage trucks entering a particular waste-management facility are weighed prior to offloading their contents. Let \({\rm{X = }}\)the total processing time for a randomly selected truck at this facility (waiting, weighing, and offloading). The article "Estimating Waste Transfer Station Delays Using GPS" (Waste Mgmt., \({\rm{2008: 1742 - 1750}}\)) suggests the plausibility of a normal distribution with mean \({\rm{13\;min}}\)and standard deviation \({\rm{4\;min}}\)for\({\rm{X}}\). Assume that this is in fact the correct distribution.

a. What is the probability that a single truck's processing time is between \({\rm{12}}\) and \({\rm{15\;min}}\)?

b. Consider a random sample of \({\rm{16}}\) trucks. What is the probability that the sample mean processing time is between \({\rm{12}}\) and\({\rm{15\;min}}\)?

c. Why is the probability in (b) much larger than the probability in (a)?

d. What is the probability that the sample mean processing time for a random sample of \({\rm{16}}\) trucks will be at least\({\rm{20\;min}}\)?

A stockroom currently has \({\rm{30}}\) components of a certain type, of which \({\rm{8}}\) were provided by supplier \({\rm{1,10}}\)by supplier \({\rm{2}}\) , and \({\rm{12}}\) by supplier \({\rm{3}}\). Six of these are to be randomly selected for a particular assembly. Let \({\rm{X = }}\) the number of supplier l's components selected, \({\rm{Y = }}\) the number of supplier \({\rm{2}}\) 's components selected, and \({\rm{p(x,y)}}\) denote the joint pmf of \({\rm{X}}\) and\({\rm{Y}}\).

a. What is \({\rm{p(3,2)}}\) ? (Hint: Each sample of size \({\rm{6}}\) is equally likely to be selected. Therefore, \({\rm{p(3,2) = }}\) (number of outcomes with \({\rm{X = 3}}\) and \({\rm{Y = 2)/(}}\) total number of outcomes). Now use the product rule for counting to obtain the numerator and denominator.)

b. Using the logic of part (a), obtain \({\rm{p(x,y}}\) ). (This can be thought of as a multivariate hypergeometric distribution-sampling without replacement from a finite population consisting of more than two categories.)

Answer

Each front tire on a particular type of vehicle is supposed to be filled to a pressure of 26 psi. Suppose the actual air pressure in each tire is a random variable—X for the right tire and Y for the left tire, with joint pdf fsx, yd 5 5 Ksx2 1 y2 d 20 # x # 30, 20 # y # 30 0 otherwise

\({{\rm{f}}_{\rm{X}}}{\rm{(x) = }}\left\{ {\begin{array}{*{20}{l}}{{\rm{K(}}{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}{\rm{)}}}&{,{\rm{20}} \le {\rm{x}} \le {\rm{30,20}} \le {\rm{y}} \le {\rm{30}}}\\{\rm{0}}&{,{\rm{ otherwise }}}\end{array}} \right.\)

a. What is the value of K?

b. What is the probability that both tires are underfilled?

c. What is the probability that the difference in air pressure between the two tires is at most 2 psi?

d. Determine the (marginal) distribution of air pressure in the right tire alone.

e. Are X and Y independent rv’s?z