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Chapter 5: Q23E (page 219)

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.

Short Answer

Expert verified

The expected difference is \({\rm{E}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{ - }}{{\rm{X}}_{\rm{2}}}} \right){\rm{ = 0}}{\rm{.15}}{\rm{.}}\)

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

Step 2: Calculating the expected difference

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 \({\rm{E(g(X,Y)) = }}\left\{ {\begin{aligned}{*{20}{l}}{\sum\limits_{\rm{x}} {\sum\limits_{\rm{y}} {\rm{g}} } {\rm{(x,y) \times p(x,y)}}\;\;\;{\rm{,X and Y discrete }}}\\{\int_{{\rm{ - 楼}}}^{\rm{楼}} {\int_{{\rm{ - 楼}}}^{\rm{楼}} {\rm{g}} } {\rm{(x,y) \times f(x,y)dxdy}}\;\;\;{\rm{,X and Y continuous}}{\rm{. }}}\end{aligned}} \right.\)

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

Therefore, since \({{\rm{X}}_{\rm{1}}}\) and \({{\rm{X}}_{\rm{2}}}\) are discrete, the following is true

\(\begin{aligned}{\rm{E}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{ - }}{{\rm{X}}_{\rm{2}}}} \right){\rm{ = }}\sum\limits_{\rm{x}} {\sum\limits_{\rm{y}} {\rm{g}} } {\rm{(x,y) \times p(x,y)}}\\{\rm{ = (0 - 0) \times 0}}{\rm{.08 + (0 - 1) \times 0}}{\rm{.07 + \ldots + (4 - 2) \times 0}}{\rm{.05 + (4 - 3) \times 0}}{\rm{.06}}\\{\rm{ = 0}}{\rm{.15}}\end{aligned}\)

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Most popular questions from this chapter

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

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

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?

There are \({\rm{40}}\) students in an elementary statistics class. On the basis of years of experience, the instructor knows that the time needed to grade a randomly chosen first examination paper is a random variable with an expected value of \({\rm{6}}\)min and a standard deviation of \({\rm{6}}\)min.

a. If grading times are independent and the instructor begins grading at \({\rm{6:50}}\) p.m. and grades continuously, what is the (approximate) probability that he is through grading before the \({\rm{11:00}}\) p.m. TV news begins?

b. If the sports report begins at \({\rm{11:10,}}\) what is the probability that he misses part of the report if he waits until grading is done before turning on the TV?

Suppose the amount of liquid dispensed by a certain machine is uniformly distributed with lower limit \({\rm{A = 8oz}}\) and upper limit\({\rm{B = 10oz}}\). Describe how you would carry out simulation experiments to compare the sampling distribution of the (sample) fourth spread for sample sizes\({\rm{n = 5,10,20}}\), and\({\rm{30}}\).
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