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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

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?

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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.

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