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Chapter 3: Q10E (page 99)

The number of pumps in use at both a six-pump station and a four-pump station will be determined. Give the possible values for each of the following random variables:

a. T = the total number of pumps in use.

b. X = the difference between the numbers in use atstations 1 and 2.

c. U = the maximum number of pumps in use at either Station.

d. Z = the number of stations having exactly two pumps in use.

Short Answer

Expert verified

a. The possible values are\(T = \left\{ {0,1,2,3,4,5,6,7,8,9,10} \right\}\)

b. The possible values are\(X = \left\{ { - 4, - 3, - 2, - 1,0,1,2,3,4,5,6} \right\}\)

c. The possible values are\(U = \left\{ {0,1,2,3,4,5,6} \right\}\)

d. The possible values are\(Z = \left\{ {0,1,2} \right\}\)

Step by step solution

01

Given information

The data are provided that consists of both six-pump station and four-pump station.

02

Determine the possible values of random variable T

a.

ConsiderTto be total number of pumps in use out of two pump stations. One pump station consists of 6 pump stations while the other contains 4 stations.

In total, there are 10 pump stations. So, the minimum value of random variable is 0 and maximum is 10.

Then, Tcan be defined by:

\(T = \left\{ {0,1,2,3,4,5,6,7,8,9,10} \right\}\)

03

Determine the possible values of random variable X

b.

ConsiderXto bethe difference between the used pumps in two stations.

In this case, obtain the different combinations between the two-pump stations by accompanying table.

0

1

2

3

4

5

6

0

\[0--0 = 0\]

\[1--0 = 1\]

\[2--0 = 2\]

\[3--0 = 3\]

\[4--0 = 4\]

\[5--0 = 5\]

\[6--0 = 6\]

1

\[0--1 = - 1\]

\[1--1 = 0\]

\[2--1 = 1\]

\[3--1 = 2\]

\[4--1 = 3\]

\[5--1 = 4\]

\[6--1 = 5\]

2

\[0--2 = - 2\]

\[1--2 = - 1\]

\[2--2 = 0\]

\[3--2 = 1\]

\[4--2 = 2\]

\[5--2 = 3\]

\[6--2 = 4\]

3

\[0--3 = - 3\]

\[1--3 = - 2\]

\[2--3 = - 1\]

\[3--3 = 0\]

\[4--3 = 1\]

\[5--3 = 2\]

\[6--3 = 3\]

4

\[0--4 = - 4\]

\[1--4 = - 3\]

\(2 - 4 = - 2\)

\[3--4 = - 1\]

\[4--4 = 0\]

\[5--4 = 1\]

\[6--4 = 2\]

The extreme points are -4 and 6. Then, X can be defined by:

\(X = \left\{ { - 4, - 3, - 2, - 1,0,1,2,3,4,5,6} \right\}\)

04

Determine the possible values of random variable U

c.

Consider U to be the maximum number of pumps in use in either of the two stations.

In this case, obtain the different combinations between the two-pump stations by accompanying table.

0

1

2

3

0

\[\max \left( {0,0} \right) = 0\]

\[\max \left( {1,0} \right) = 1\]

\[\max \left( {2,0} \right) = 2\]

\[\max \left( {3,0} \right) = 3\]

1

\[\max \left( {0,1} \right) = 1\]

\[\max \left( {1,1} \right) = 1\]

\[\max \left( {2,1} \right) = 2\]

\[\max \left( {3,1} \right) = 3\]

2

\[\max \left( {0,2} \right) = 2\]

\[\max \left( {1,2} \right) = 2\]

\[\max \left( {2,2} \right) = 2\]

\[\max \left( {3,2} \right) = 3\]

3

\[\max \left( {0,3} \right) = 3\]

\[\max \left( {1,3} \right) = 3\]

\[\max \left( {2,3} \right) = 3\]

\[\max \left( {3,3} \right) = 3\]

4

\[\max \left( {0,4} \right) = 4\]

\[\max \left( {1,4} \right) = 4\]

\[\max \left( {2,4} \right) = 4\]

\[\max \left( {3,4} \right) = 4\]

4

5

6

0

\[\max \left( {4,0} \right) = 4\]

\[\max \left( {5,0} \right) = 5\]

\[\max \left( {6,0} \right) = 6\]

1

\[\max \left( {4,1} \right) = 4\]

\[\max \left( {5,1} \right) = 5\]

\[\max \left( {6,1} \right) = 6\]

2

\[\max \left( {4,2} \right) = 4\]

\[\max \left( {5,2} \right) = 5\]

\[\max \left( {6,2} \right) = 6\]

3

\[\max \left( {4,3} \right) = 4\]

\[\max \left( {5,3} \right) = 5\]

\[\max \left( {6,3} \right) = 6\]

4

\[\max \left( {4,4} \right) = 4\]

\[\max \left( {5,4} \right) = 5\]

\[\max \left( {6,4} \right) = 6\]

The extreme points are 0 and 6. Then, Ucan be defined by:

\(U = \left\{ {0,1,2,3,4,5,6} \right\}\)

05

Determine the possible values of random variable Z

d.

ConsiderZto be the number of stations having exactly two pumps in use in two stations.

In this case, obtain the different combinations between the two-pump stations by accompanying table.

0

1

2

3

4

5

6

0

\[\left( {0 + 0} \right) = 0\]

\[\left( {1 + 0} \right) = 1\]

\[\left( {2 + 0} \right) = 2\]

\[\left( {3 + 0} \right) = 3\]

\[\left( {4 + 0} \right) = 4\]

\[\left( {5 + 0} \right) = 5\]

\[\left( {6 + 0} \right) = 6\]

1

\[\left( {0 + 1} \right) = 1\]

\[\left( {1 + 1} \right) = 2\]

\[\left( {2 + 1} \right) = 3\]

\[\left( {3 + 1} \right) = 4\]

\[\left( {4 + 1} \right) = 5\]

\[\left( {5 + 1} \right) = 6\]

\[\left( {6 + 1} \right) = 7\]

2

\[\left( {0 + 2} \right) = 2\]

\[\left( {1 + 2} \right) = 3\]

\[\left( {2 + 2} \right) = 4\]

\[\left( {3 + 2} \right) = 5\]

\[\left( {4 + 2} \right) = 6\]

\[\left( {5 + 2} \right) = 7\]

\[\left( {6 + 2} \right) = 8\]

3

\[\left( {0 + 3} \right) = 3\]

\[\left( {1 + 3} \right) = 4\]

\[\left( {2 + 3} \right) = 5\]

\[\left( {3 + 3} \right) = 6\]

\[\left( {4 + 3} \right) = 7\]

\[\left( {5 + 3} \right) = 8\]

\[\left( {6 + 3} \right) = 9\]

4

\[\left( {0 + 4} \right) = 4\]

\[\left( {1 + 4} \right) = 5\]

\[\left( {2 + 4} \right) = 6\]

\[\left( {3 + 4} \right) = 7\]

\[\left( {4 + 4} \right) = 8\]

\[\left( {5 + 4} \right) = 9\]

\[\left( {6 + 0} \right) = 6\]

The minimum and maximum numbers are 0 and 10 respectively.

Then,\(Z\)can be defined by:

\(Z = \left\{ {0,1,2,3,4,6,7,8,9,10} \right\}\)

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