(Solution) - With the probabilities as given as in Section 7 1 Exercise -(2025 Original AI-Free Solution)

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With the probabilities as given (as in Section 7.1, Exercise 33).
Recall the cells in Section 6.4, Exercises 31 and 32. New cells stain properly with probability 0.95, 1-day-old cells stain properly with probability 0.9, 2-day-old cells stain properly with probability 0.8, and 3-day-old cells stain properly with probability 0.5. Suppose
Pr(cell is 0 days old) = 0.4
Pr(cell is 1 day old) = 0.3
Pr(cell is 2 days old) = 0.2
Pr(cell is 3 days old) = 0.1.
Using the random variables A for age and S for staining, find the covariance in the following cases. Explain why the covariance is positive or negative.