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Chapter 5: Q26E (page 219)

Consider a small ferry that can accommodate cars and buses. The toll for cars is\({\rm{\$ 3}}\), and the toll for buses is\({\rm{\$ 10}}\). Let \({\rm{X}}\) and \({\rm{Y}}\) denote the number of cars and buses, respectively, carried on a single trip. Suppose the joint distribution of \({\rm{X}}\) and\({\rm{Y}}\). Compute the expected revenue from a single trip.

Short Answer

Expert verified

The expected revenue is\({\rm{E(3X + 10Y) = 15}}{\rm{.4}}\)

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

Calculating expected revenue

The

Expected Value

(Mean value) of a random variable\({\rm{g(X,Y)}}\), where \({\rm{g( \times )}}\) is a function, denoted as \({\rm{E(g(X,Y))}}\) is given by

\(E(g(X,Y)) = \left\{ {\begin{aligned}{*{20}{l}}{\sum\limits_x {\sum\limits_y g } (x,y) \cdot p(x,y)}&{,X{\rm{ and }}Y{\rm{ discrete }}}\\{\int_{ - \infty }^\infty {\int_{ - \infty }^\infty g } (x,y) \cdot f(x,y)dxdy}&{,X{\rm{ and }}Y{\rm{ continuous}}{\rm{. }}}\end{aligned}} \right.\)

where \({\rm{p(x,y)}}\) is pmf and \({\rm{f(x,y)}}\) pdf.

We are given pmf, therefore we can compute the expected revenue from a single trip, where \({\rm{g(X,Y) = 3X + 10Y(3}}\) dollars for cars and \({\rm{10}}\) dollars for buses). The following is true

\(\begin{aligned}{\rm{E(3X + 10Y) = }}\sum\limits_{\rm{x}} {\sum\limits_{\rm{y}} {{\rm{(3x + 10y)}}} } {\rm{ \times p(x,y)}}\\{\rm{ = (3 \times 0 + 10 \times 0) \times p(0,0) + (3 \times 0 + 10 \times 1) \times p(0,1) + \ldots + (3 \times 5 + 10 \times 2) \times p(5,2)}}\\{\rm{ = 0 \times 0}}{\rm{.025 + 10 \times 0}}{\rm{.015 + \ldots + 35 \times 0}}{\rm{.02}}\\{\rm{ = 15}}{\rm{.4}}{\rm{.}}\end{aligned}\)

Therefore, the expected revenue is\({\rm{E(3X + 10Y) = 15}}{\rm{.4}}\).

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Most popular questions from this chapter

Answer the following questions:

a. Given that\({\rm{X = 1}}\), determine the conditional pmf of \({\rm{Y}}\)-i.e., \({{\rm{p}}_{{\rm{Y}}\mid {\rm{X}}}}{\rm{(0}}\mid {\rm{1),}}{{\rm{p}}_{{\rm{Y}}\mid {\rm{X}}}}{\rm{(1}}\mid {\rm{1)}}\), and\({{\rm{p}}_{{\rm{Y}}\mid {\rm{X}}}}{\rm{(2}}\mid {\rm{1)}}\).

b. Given that two houses are in use at the self-service island, what is the conditional pmf of the number of hoses in use on the full-service island?

c. Use the result of part (b) to calculate the conditional probability\({\rm{P(Y拢

1}}\mid {\rm{X = 2)}}\).

d. Given that two houses are in use at the full-service island, what is the conditional pmf of the number in use at the self-service island?

I have three errands to take care of in the Administration Building. Let \({\rm{Xi = }}\)the time that it takes for the \({\rm{ ith}}\)errand\({\rm{(i = 1,2,3)}}\), and let \({{\rm{X}}_{\rm{4}}}{\rm{ = }}\)the total time in minutes that I spend walking to and from the building and between each errand. Suppose the \({\rm{Xi`s}}\)are independent, and normally distributed, with the following means and standard deviations: \({{\rm{\mu }}_{{{\rm{X}}_{\rm{1}}}}}{\rm{ = 15,}}{{\rm{\sigma }}_{{{\rm{X}}_{\rm{1}}}}}{\rm{ = 4,}}{{\rm{\mu }}_{{{\rm{X}}_{\rm{2}}}}}{\rm{ = 5,}}{{\rm{\sigma }}_{{{\rm{X}}_{\rm{2}}}}}{\rm{ = 1,}}{{\rm{\mu }}_{{{\rm{X}}_{\rm{3}}}}}{\rm{ = 8,}}{{\rm{\sigma }}_{{{\rm{X}}_{\rm{3}}}}}{\rm{ = 2,}}{{\rm{\mu }}_{{{\rm{X}}_{\rm{4}}}}}{\rm{ = 12,}}{{\rm{\sigma }}_{{{\rm{X}}_{\rm{4}}}}}{\rm{ = 3}}\).plan to leave my office at precisely \({\rm{10:00}}\)a.m. and wish to post a note on my door that reads, 鈥淚 will return by \({\rm{t}}\)a.m.鈥� What time\({\rm{t}}\)should I write down if I want the probability of my arriving after \({\rm{t}}\) to be \({\rm{.01}}\)?

Six individuals, including \({\rm{A}}\) and \({\rm{B}}\), take seats around a circular table in a completely random fashion. Suppose the seats are numbered\({\rm{1, \ldots ,6}}\). Let \({\rm{X = }}\) A's seat number and \({\rm{Y = B}}\) 's seat number. If A sends a written message around the table to \({\rm{B}}\) in the direction in which they are closest, how many individuals (including A and B) would you expect to handle the message?

Let\({{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}{\rm{,}}{{\rm{X}}_{\rm{3}}}{\rm{,}}{{\rm{X}}_{\rm{4}}}{\rm{,}}{{\rm{X}}_{\rm{5}}}\), and \({{\rm{X}}_{\rm{6}}}\) denote the numbers of blue, brown, green, orange, red, and yellow M\&M candies, respectively, in a sample of size\({\rm{n}}\). Then these \({{\rm{X}}_{\rm{i}}}\) 's have a multinomial distribution. According to the M\&M Web site, the color proportions are\({{\rm{p}}_{\rm{1}}}{\rm{ = }}{\rm{.24,}}{{\rm{p}}_{\rm{2}}}{\rm{ = }}{\rm{.13}}\), \({{\rm{p}}_{\rm{3}}}{\rm{ = }}{\rm{.16,}}{{\rm{p}}_{\rm{4}}}{\rm{ = }}{\rm{.20,}}{{\rm{p}}_{\rm{5}}}{\rm{ = }}{\rm{.13}}\), and\({{\rm{p}}_{\rm{6}}}{\rm{ = }}{\rm{.14}}\).

a. If\({\rm{n = 12}}\), what is the probability that there are exactly two M\&Ms of each color?

b. For\({\rm{n = 20}}\), what is the probability that there are at most five orange candies? (Hint: Think of an orange candy as a success and any other color as a failure.)

c. In a sample of\({\rm{20M \backslash Ms}}\), what is the probability that the number of candies that are blue, green, or orange is at least \({\rm{10}}\) ?

Two different professors have just submitted final exams for duplication. Let \({\rm{X}}\) denote the number of typographical errors on the first professor鈥檚 exam and \({\rm{Y}}\) denote the number of such errors on the second exam. Suppose \({\rm{X}}\) has a Poisson distribution with parameter \({{\rm{\mu }}_{\rm{1}}}\), \({\rm{Y}}\) has a Poisson distribution with parameter \({{\rm{\mu }}_{\rm{2}}}\), and \({\rm{X}}\) and \({\rm{Y}}\) are independent.

a. What is the joint pmf of \({\rm{X}}\) and\({\rm{Y}}\)?

b. What is the probability that at most one error is made on both exams combined?

c. Obtain a general expression for the probability that the total number of errors in the two exams is m (where \({\rm{m}}\) is a nonnegative integer). (Hint: \({\rm{A = }}\left\{ {\left( {{\rm{x,y}}} \right){\rm{:x + y = m}}} \right\}{\rm{ = }}\left\{ {\left( {{\rm{m,0}}} \right)\left( {{\rm{m - 1,1}}} \right){\rm{,}}.....{\rm{(1,m - 1),(0,m)}}} \right\}\)Now sum the joint pmf over \({\rm{(x,y)}} \in {\rm{A}}\)and use the binomial theorem, which says that

\({\rm{P(X + Y = m)}}\mathop {\rm{ = }}\limits^{{\rm{(1)}}} {\sum\limits_{{\rm{k = 0}}}^{\rm{m}} {\left( {\begin{array}{*{20}{c}}{\rm{m}}\\{\rm{k}}\end{array}} \right){{\rm{a}}^{\rm{k}}}{{\rm{b}}^{{\rm{m - k}}}}{\rm{ = }}\left( {{\rm{a + b}}} \right)} ^{\rm{m}}}\)
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