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Chapter 3: Problem 10

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 at stations 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: 0-10; b: -4 to 6; c: 0-6; d: 0, 1, 2

Step by step solution

01

Analyze Total Pumps in Use (T)

The total number of pumps in use, denoted by \(T\), is the sum of the pumps in use at both stations. The first station can have 0 to 6 pumps in use, and the second station can have 0 to 4 pumps in use. Therefore, \(T\) can range from \(0 + 0 = 0\) to \(6 + 4 = 10\). Thus, the possible values for \(T\) are \(0, 1, 2, \ldots, 10\).
02

Analyze Difference in Pumps in Use (X)

The number \(X\) is the difference between the pumps in use at the first station and the second. If \(X = n_1 - n_2\), where \(n_1\) is between 0 and 6 and \(n_2\) is between 0 and 4, then \(X\) can vary from \(-(4) = -4\) to \(6 - 0 = 6\). Thus, the possible values for \(X\) are \(-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6\).
03

Analyze Maximum Pumps in Use (U)

\(U\) denotes the maximum number of pumps in use at either station. The maximum possible number can be either from station 1 with 6 pumps or station 2 with 4 pumps. Thus, \(U\) ranges from a minimum of 0 (if both stations have 0 in use) to a maximum of 6 (if the first station has all pumps in use). The possible values are \(0, 1, 2, 3, 4, 5, 6\).
04

Analyze Stations with Exactly Two Pumps in Use (Z)

\(Z\) represents the number of stations with exactly two pumps in use. The possible number of stations that can have exactly two pumps in use are 0, 1, or 2. Each station can either have two pumps in use or not (0), and there are two stations to consider. Thus, the possible values for \(Z\) are \(0, 1, 2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Total Number of Pumps
The total number of pumps in use is represented by the variable \(T\). To find the potential values for \(T\), we need to consider both fueling stations in the scenario. One station has a maximum capacity of six pumps, while the other has four pumps.
This means that if all pumps at both stations are actively being used at the same time, the maximum total is 6 (from the first station) plus 4 (from the second station), which equals 10 pumps in use. Conversely, if no pumps are active, the total would be 0. Thus, the range of possible values for \(T\) stretches from 0 to 10.
  • A minimum of 0 pumps when both stations are not in use.
  • A maximum of 10 pumps when all pumps at both stations are in use.
The values are continuous, so \(T\) can be any integer between 0 and 10 inclusive.
Difference in Pumps
The difference in the number of pumps in use between the two stations is denoted by \(X\). We calculate this by taking the number of pumps in use at the first station and subtracting the number in use at the second.
For instance, if the first station uses 3 pumps and the second uses 1, the result is \(3 - 1 = 2\). This means the difference can vary significantly based on the distribution of use between the two stations.
  • Maximum positive difference: If station 1 has all 6 pumps in use and station 2 has none, \(X = 6 - 0 = 6\).
  • Maximum negative difference: If station 1 has no pumps in use and station 2 has all 4, \(X = 0 - 4 = -4\).
  • Neutral difference: Both stations in use or not in use, or equal numbers in use, result in \(X = 0\).
Therefore, \(X\) could take on any integer value from -4 to 6, depicting a range of possible differences.
Maximum Pumps in Use
The maximum number of pumps in use at either station is represented by \(U\). Essentially, \(U\) is derived by comparing the number of pumps in use between both stations and selecting the larger value.
For example, if one station uses 5 pumps and the other uses 3, the maximum is 5, so \(U = 5\). This concept identifies how many pumps are concurrently in serious demand at the busiest station.
  • The least number for \(U\) is 0, when no pumps are in use at either station.
  • The greatest possible number is 6, when all pumps at station 1 are active.
This means \(U\) spans from 0 to 6 with incremental integers, ensuring that all potential scenarios are covered.
Number of Stations with Two Pumps
The number of stations with exactly two pumps in use is denoted by \(Z\). This variable measures how many of the available stations are using precisely two pumps at any given time.
It is possible for:
  • None of the stations to use exactly two pumps, meaning \(Z = 0\).
  • One station to have exactly two pumps in operation, resulting in \(Z = 1\).
  • Both stations to use exactly two pumps, implying \(Z = 2\).
Hence, \(Z\) can take values from 0 to 2. It provides insight into the utilization pattern, focusing on the specific scenario where stations operate with an intermediate level of pump usage.

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