Final answer:
The CDF in this question represents the amount of time a book on reserve is checked out. To find the probabilities requested, we can use the CDF. For example, P(X > 2) can be found by subtracting P(X < 2) from 1. Similarly, the probability a patron will check out at least one book is 1, and the probability of taking out no more than three books is 9.
Step-by-step explanation:
Let X denote the amount of time a book on two-hour reserve is actually checked out, and suppose the cumulative distribution function (CDF) is given by:
F(x) = 0 x<0 0 < x < 4 x² 4 < x
To find P(X > 2), we can use the complement rule. P(X > 2) = 1 - P(X < 2). Substituting into the CDF gives P(X < 2) = 0. So, P(X > 2) = 1 - 0 = 1.
The probability that a patron will check out at least one book can be found by subtracting the probability of not checking out any books from 1. The probability of not checking out any books is P(X < 0) = 0. So, P(at least one book) = 1 - 0 = 1.
To find the probability that a patron will take out no more than three books, we need to find P(X <= 3). This can be found by evaluating the CDF at 3: P(X <= 3) = 3 <sup>2 = 9.