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Chapter 5: Q3E (page 210)

A certain market has both an express checkout line and a superexpress checkout line. Let \({{\rm{X}}_{\rm{1}}}\) denote the number of customers in line at the express checkout at a particular time of day, and let \({{\rm{X}}_{\rm{2}}}\) denote the number of customers in line at the superexpress checkout at the same time. Suppose the joint pmf of \({{\rm{X}}_{\rm{1}}}\)and\({{\rm{X}}_{\rm{2}}}\) is as given in the accompanying table

a. What is \({\rm{P(}}{{\rm{X}}_{\rm{1}}}{\rm{ = 1}}{{\rm{X}}_{\rm{2}}} = 1)\), that is, the probability that there is exactly one customer in each line?

b. What is \({\rm{P(}}{{\rm{X}}_{\rm{1}}}{\rm{ = }}{{\rm{X}}_{\rm{2}}})\),that is, the probability that the numbers of customers in the two lines are identical? c. Let A denote the event that there are at least two more customers in one line than in the other line. Express A in terms of \({{\rm{X}}_{\rm{1}}}\) and \({{\rm{X}}_{\rm{2}}}\), and calculate the probability of this event.

d. What is the probability that the total number of customers in the two lines is exactly four? At least four?

Short Answer

Expert verified
  1. The probability is \({\rm{P}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{ = 1,}}{{\rm{X}}_{\rm{2}}}{\rm{ = 1}}} \right){\rm{ = 0}}{\rm{.15;}}\)
  2. The probability is \({\rm{ P}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{ = }}{{\rm{X}}_{\rm{2}}}} \right){\rm{ = 0}}{\rm{.4;}}\)
  3. The probability is \({\rm{ P(A) = 0}}{\rm{.22;}}\)
  4. The probability is \({\rm{P}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{ + }}{{\rm{X}}_{\rm{2}}} \ge {\rm{4}}} \right){\rm{ = 0}}{\rm{.46}}{\rm{.}}\)

Step by step solution

01

Definition of Probability

Probability is a metric for determining the possibility of an event occurring. Many things are impossible to forecast with\({\rm{100\% }}\)accuracy. Using it, we can only anticipate the probability of an event occurring, or how probable it is to occur. Probability can range from\({\rm{0}}\)to\({\rm{1}}\), with\({\rm{0}}\)indicating an improbable event and 1 indicating a certain event. Possibility of...

02

Find the probability that there is exactly one customer in each line?

(a):

As an entry, the following probability can be read from the given table.\((1,1)\)-first column/first row

\(\begin{aligned}{\rm{P}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{ = 1,}}{{\rm{X}}_{\rm{2}}}{\rm{ = 1}}} \right)\\{\rm{ = p(1,1)}}\\{\rm{ = 0}}{\rm{.15}}{\rm{.}}\end{aligned}\)

Therefore , the probability is \( = 0.15\).

03

 Find the probability that the numbers of customers?

(b):

The following holds

\(\begin{aligned}{\rm{P}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{ = }}{{\rm{X}}_{\rm{2}}}} \right){\rm{ = p(0,0) + p(1,1) + p(2,2) + p(3,3)}}\\{\rm{ = 0}}{\rm{.08 + 0}}{\rm{.15 + 0}}{\rm{.1 + 0}}{\rm{.7}}\\{\rm{ = 0}}{\rm{.4}}\end{aligned}\)

Therefore , the probability is \( = 0.4\).

04

Calculate the probability of this event?

(c):

The following two events can be combined to form the provided event.

\(\left\{ {{{\rm{X}}_{\rm{1}}} \ge {\rm{2 + }}{{\rm{X}}_{\rm{2}}}} \right\}{\rm{ and }}\left\{ {{{\rm{X}}_{\rm{2}}} \ge 2 + {{\rm{X}}_{\rm{1}}}} \right\}.\)

Therefore, we have

\({\rm{A = }}\left\{ {\left\{ {{{\rm{X}}_{\rm{1}}} \ge 2 + {{\rm{X}}_{\rm{2}}}} \right\} \cup \left\{ {{{\rm{X}}_{\rm{2}}} \ge 2 + {{\rm{X}}_{\rm{1}}}} \right\}} \right\}\)

The event \({\rm{A}}\)can be represented as the sum of nine discrete occurrences in form.

\(\left\{ {{{\rm{X}}_{\rm{1}}}{\rm{ = }}{{\rm{x}}_{\rm{1}}}} \right\}{\rm{,}}\left\{ {{{\rm{X}}_{\rm{2}}}{\rm{ = }}{{\rm{x}}_{\rm{2}}}} \right\}\)

where \({{\rm{x}}_{\rm{1}}}{\rm{,}}{{\rm{x}}_{\rm{2}}}\) satisfy the event \({\rm{A}}\)'s specified property. As a result, the chance can be computed as follows:

\(\begin{aligned}{\rm{P(A) = P}}\left( {\left\{ {{X_1} \ge 2 + {X_2}} \right\} \cup \left\{ {{X_2} \ge 2 + {X_1}} \right\}} \right)\\{\rm{ = p(2,0) + p(3,0) + p(4,0) + p(3,1) + p(4,1) + p(4,2) + p(0,2) + p(0,3) + p(1,3)}}\\{\rm{ = 0}}{\rm{.05 + 0}}{\rm{.00 + 0}}{\rm{.00 + 0}}{\rm{.03 + 0}}{\rm{.01 + 0}}{\rm{.05 + 0}}{\rm{.04 + 0}}{\rm{.00 + 0}}{\rm{.04}}\\{\rm{ = 0}}{\rm{.22}}{\rm{.}}\end{aligned}\)

Therefore , the probability is \({\rm{ = }}{\rm{.22}}\).

05

Find the probability that the total number of customers?

(d):

The following holds

\(\begin{aligned}{\rm{P}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{ + }}{{\rm{X}}_{\rm{2}}}{\rm{ = 4}}} \right){\rm{ = p(1,3) + p(2,2) + p(3,1) + p(4,0)}}\\{\rm{ = 0}}{\rm{.04 + 0}}{\rm{.1 + 0}}{\rm{.03 + 0}}{\rm{.00}}\\{\rm{ = 0}}{\rm{.17}}{\rm{.}}\end{aligned}\)

And for the second part the following is true

\(\begin{aligned}{l}{\rm{P}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{ + }}{{\rm{X}}_{\rm{2}}} \ge {\rm{4}}} \right){\rm{ = p(1,3) + p(2,2) + p(3,1) + p(4,0) + p(4,1) + p(4,2) + p(4,3) + p(3,2) + p(3,3) + p(2,3)}}\\{\rm{ = 0}}{\rm{.46}}\end{aligned}\)

All values can be found at the given table (see \({\rm{(a)}}\)).

Therefore , the probability is \({\rm{ = 0}}{\rm{.46}}\).

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Most popular questions from this chapter

A service station has both self-service and full-service islands. On each island, there is a single regular unleaded pump with two hoses. Let \({\rm{X}}\)denote the number of hoses being used on the self-service island at a particular time, and let\({\rm{Y}}\)denote the number of hoses on the full-service island in use at that time. The joint \({\rm{pmf}}\) of \({\rm{X}}\)and \({\rm{Y}}\) appears in the accompanying tabulation.

a. What is\({\rm{P(X = 1 and Y = 1)}}\)?

b. Compute P(X拢1}and{Y拢1)

c. Give a word description of the event , and compute the probability of this event.

d. Compute the marginal \({\rm{pmf}}\) of \({\rm{X}}\)and of \({\rm{Y}}\). Using \({{\rm{p}}_{\rm{X}}}{\rm{(x)}}\)what is P(X拢1)?

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

Consider a system consisting of three components as pictured. The system will continue to function as long as the first component functions and either component \({\rm{2}}\) or component \({\rm{3}}\)functions. Let \({{\rm{X}}_{{\rm{1,}}}}{{\rm{X}}_{\rm{2}}}\), and \({{\rm{X}}_{\rm{3}}}\) denote the lifetimes of components \({\rm{1}}\), \({\rm{2}}\), and \({\rm{3}}\), respectively. Suppose the \({{\rm{X}}_{\rm{i}}}\) 鈥檚 are independent of one another and each \({{\rm{X}}_{\rm{i}}}\) has an exponential distribution with parameter \({\rm{\lambda }}\).

a. Let \({\rm{Y}}\) denote the system lifetime. Obtain the cumulative distribution function of \({\rm{Y}}\)and differentiate to obtain the pdf. (Hint: \({{\rm{F}}_{\left( {\rm{Y}} \right)}}{\rm{P}}\left\{ {{\rm{Y}} \le {\rm{y}}} \right\}\); express the event \(\left\{ {{\rm{Y}} \le {\rm{y}}} \right\}\)in terms of unions and/or intersections of the three events \(\left\{ {{{\rm{X}}_{\rm{i}}} \le {\rm{y}}} \right\}\), \(\left\{ {{{\rm{X}}_{\rm{2}}} \le {\rm{y}}} \right\}\), and \(\left\{ {{{\rm{X}}_3} \le {\rm{y}}} \right\}\).)

b. Compute the expected system lifetime

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鈥檚? Explain.

Suppose the sediment density (g/cm) of a randomly selected specimen from a certain region is normally distributed with mean \({\rm{2}}{\rm{.65 }}\)and standard deviation \({\rm{.85}}\) (suggested in 鈥淢odeling Sediment and Water Column Interactions for Hydrophobic Pollutants,鈥� Water Research, \({\rm{1984: 1164 - 1174 }}\)).

a. If a random sample of \({\rm{25}}\)specimens is selected, what is the probability that the sample average sediment density is at most \({\rm{3}}{\rm{.00 }}\)? Between \({\rm{2}}{\rm{.65 }}\)and \({\rm{3}}{\rm{.00 }}\)?

b. How large a sample size would be required to ensure that the first probability in part (a) is at least \({\rm{.99}}\)?

Suppose the expected tensile strength of type-A steel is \({\rm{105ksi}}\)and the standard deviation of tensile strength is \({\rm{8ksi}}\). For type-B steel, suppose the expected tensile strength and standard deviation of tensile strength are \({\rm{100ksi}}\)and \({\rm{6ksi}}\), respectively. Let \({\rm{\bar X = }}\)the sample average tensile strength of a random sample of \({\rm{40}}\) type-A specimens, and let \({\rm{\bar Y = }}\)the sample average tensile strength of a random sample of \({\rm{35}}\)type-B specimens.

a. What is the approximate distribution of \({\rm{\bar X ? of \bar Y?}}\)

b. What is the approximate distribution of \({\rm{\bar X - \bar Y}}\)? Justify your answer.

c. Calculate (approximately) \(P( - 1拢\bar X - \bar Y拢1)\)

d. Calculate. If you actually observed , would you doubt that \({{\rm{\mu }}_{\rm{1}}}{\rm{ - }}{{\rm{\mu }}_{\rm{2}}}{\rm{ = 5?}}\)
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