EXAMPLE 4.32
How many courses? In Exercise 4.45 (page 244) you described the probability distribution of the number of courses taken in the fall by students at a small liberal arts college. Here is the distribution:
| Courses in the fall | 1 | 2 | 3 | 4 | 5 | 6 |
| Probability | 0.05 | 0.05 | 0.13 | 0.26 | 0.36 | 0.15 |
For the spring semester, the distribution is a little different.
| Courses in the spring | 1 | 2 | 3 | 4 | 5 | 6 |
| Probability | 0.06 | 0.08 | 0.15 | 0.25 | 0.34 | 0.12 |
For a randomly selected student, let X be the number of courses taken in the fall semester, and let Y be the number of courses taken in the spring semester. The means of these random variables are
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μX = (1)(0.05) + (2)(0.05) + (3)(0.13) + (4)(0.26) + (5)(0.36) + (6)(0.15)
= 4.28
μY = (1)(0.06) + (2)(0.08) + (3)(0.15) + (4)(0.25) + (5)(0.34) + (6)(0.12)
= 4.09
The mean course load for the fall is 4.28 courses, and the mean course load for the spring is 4.09 courses. We assume that these distributions apply to students who earned credit for courses taken in the fall and the spring semesters. The mean of the total number of courses taken for the academic year is X + Y. Using Rule 2, we calculate the mean of the total number of courses:
μZ = μX + μY
= 4.28 + 4.09 = 8.37
Note that it is not possible for a student to take 8.37 courses in an academic year. This number is the mean of the probability distribution.