The number of pumps in use at both a sixpump station and a f

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. (Enter your answers as a comma-separated list.)

(a) T = the total number of pumps in use
T =

(b) X = the difference between the numbers in use at stations 1 and 2

X =

(c) U = the maximum number of pumps in use at either station

U =

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

Z =

Solution

A minimum number of pumps in use on a six-pump station could be 0.

Maximum number of pumps in use on a six-pump station would be 6.

N₆ = {0, 1, 2, 3, 4, 5, 6}

Similarly, A minimum number of pumps in use on a four-pump station could be 0 and Maximum number of pumps in use on a four-pump station would be 4.

N₄ = {0, 1, 2, 3, 4}

(a) Total number of pumps in use: T

The minimum value of total number of pumps in use = minimum number of pumps in use at six-pump station + minimum number of pumps in use at four-pump station

Thus, T_min = 0 + 0 = 0

The maximum value of total number of pumps in use = maximum number of pumps in use at six-pump station + maximum number of pumps in use at four-pump station

Thus, T_max = 6 + 4 = 10

Thus, T is a set of all numbers from 0 to 10.

Thus, T = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

(Note that T could take only integer values)

(b) X = the difference between the numbers in use at stations 1 and 2

Minimum value of a difference in the number of pumps in use at station 1 and 2 = maximum value of pumps in use - minimum value of pumps in use = 6 - 0 = 6

Minimum value of difference in the number of pumps in use at station 1 and 2 = difference between two minimum values of number of pumps in use = 0 - 0 = 0

Thus, X : {0, 1, 2, 3, 4, 5, 6}

(c) U = the maximum number of pumps in use at either station

Maximum number of pumps not in use at six-pump station is 6

Maximum number of pumps not in use at four-pump station is 4

Thus, U = {4, 6}

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

Minimum number of stations having exactly two pumps in use would be 0. i.e. neither of the station is having exactly two pumps in use.

Similarly maximum number of stations having exactly two pumps in use would be 2. i.e. both the stations are having exactly two pumps in use.

Thus, Z : {0, 1, 2}