0 0.08 0.08 0.13 0.05 0.04 0.00 0.00 0.03 0.01 0.06 The difference between the number of customers in line at the express checkout and the number in line at the superexpress checkout is X₂-X. Calculate the expected difference. 2 3 1 0.06 2 0.04 0.05 0.10 0.04 0.05 0.00 0.05 0.06 0.07

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### Calculating Expected Difference in Check-Out Line Customers

A certain market has both an express checkout line and a super-express checkout line. Let \( X_1 \) denote the number of customers in line at the express checkout at a particular time of day, and let \( X_2 \) denote the number of customers in line at the super-express checkout at the same time. Suppose the joint probability mass function (pmf) of \( X_1 \) and \( X_2 \) is given in the accompanying table.

| \(X_1 \backslash X_2\) |   0    |   1    |   2    |   3    |
|:----------------------:|:------:|:------:|:------:|:------:|
|          0             |  0.08  |  0.06  |  0.04  |  0.00  |
|          1             |  0.08  |  0.13  |  0.05  |  0.05  |
|          2             |  0.05  |  0.04  |  0.10  |  0.06  |
|          3             |  0.04  |  0.03  |  0.06  |  0.07  |
|          4             |  0.00  |  0.01  |  0.05  |  0.06  |

#### Explanation of the Table
The rows in the table represent the values of \( X_1 \) (number of customers in the express checkout line), while the columns represent the values of \( X_2 \) (number of customers in the super-express checkout line). Each cell in the table contains the joint probability \( P(X_1 = x_1, X_2 = x_2) \).

To calculate the expected difference between the number of customers in line at the express checkout and the number in line at the super-express checkout, we define \( D = X_1 - X_2 \). We then compute the expected value \( E(D) \) using the joint pmf.

#### Calculation Steps
1. Compute the expectations \( E(X_1) \) and
Transcribed Image Text:--- ### Calculating Expected Difference in Check-Out Line Customers A certain market has both an express checkout line and a super-express checkout line. Let \( X_1 \) denote the number of customers in line at the express checkout at a particular time of day, and let \( X_2 \) denote the number of customers in line at the super-express checkout at the same time. Suppose the joint probability mass function (pmf) of \( X_1 \) and \( X_2 \) is given in the accompanying table. | \(X_1 \backslash X_2\) | 0 | 1 | 2 | 3 | |:----------------------:|:------:|:------:|:------:|:------:| | 0 | 0.08 | 0.06 | 0.04 | 0.00 | | 1 | 0.08 | 0.13 | 0.05 | 0.05 | | 2 | 0.05 | 0.04 | 0.10 | 0.06 | | 3 | 0.04 | 0.03 | 0.06 | 0.07 | | 4 | 0.00 | 0.01 | 0.05 | 0.06 | #### Explanation of the Table The rows in the table represent the values of \( X_1 \) (number of customers in the express checkout line), while the columns represent the values of \( X_2 \) (number of customers in the super-express checkout line). Each cell in the table contains the joint probability \( P(X_1 = x_1, X_2 = x_2) \). To calculate the expected difference between the number of customers in line at the express checkout and the number in line at the super-express checkout, we define \( D = X_1 - X_2 \). We then compute the expected value \( E(D) \) using the joint pmf. #### Calculation Steps 1. Compute the expectations \( E(X_1) \) and
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