4.94 Two day demand.

CASE 4.2 Refer to the distribution of daily demand for blood bags in Case 4.2 (pages 210–211). Let be the total demand over two days. Assume that demand is independent from day to day.

4.95 How many courses?

At a small liberal arts college, students can register for one to six courses. In a typical fall semester, 5% take one course, 5% take two courses, 13% take three courses, 26% take four courses, 36% take five courses, and 15% take six courses. Let be the number of courses taken in the fall by a randomly selected student from this college. Describe the probability distribution of this random variable.

4.96 Make a graphical display.

Refer to the previous exercise. Use a probability histogram to provide a graphical description of the distribution of .

4.97 Find some probabilities.

Refer to Exercise 4.95.

4.98 Texas hold 'em.

The game of Texas hold 'em starts with each player receiving two cards. Here is the probability distribution for the number of aces in two-card hands:

4.99 How large are households?

Choose an American household at random, and let be the number of persons living in the household. If we ignore the few households with more than seven inhabitants, the probability model for is as follows:

4.100 How much to order?

CASE 4.2 Faced with the demand for the perishable product in blood, hospital managers need to establish an ordering policy that deals with the trade-off between shortage and wastage. As it turns out, this scenario, referred to as a single-period inventory problem, is well known in the area of operations management, and there is an optimal policy. What we need to know is the per item cost of being short and the per item cost of being in excess . In terms of the blood example, the hospital estimates that for every bag short, there is a cost of $80 per bag, which includes expediting and emergency delivery costs. Any transfusion blood bags left in excess at day's end are associated with $20 per bag cost, which includes the original cost of purchase along with end-of-day handling costs. With the objective of minimizing long-term average costs, the following critical ratio needs to be computed:

Recognize that will always be in the range of 0 to 1. It turns out that the optimal number of items to order is the smallest value of such that is at least the value.

4.101 Discrete or continuous?

In each of the following situations, decide whether the random variable is discrete or continuous, and give a reason for your answer.

4.102 Use the uniform distribution.

Suppose that a random variable follows the uniform distribution described in Example 4.26 (pages 213–214). For each of the following events, find the probability and illustrate your calculations with a sketch of the density curve similar to the ones in Figure 4.12 (page 214).

4.103 Spell-checking software.

Spell-checking software catches “nonword errors,” which are strings of letters that are not words, as when “the” is typed as “eth.” When undergraduates are asked to write a 250-word essay (without spell-checking), the number of nonword errors has the following distribution:

4.104 Find the probabilities.

Let the random variable be a random number with the uniform density curve in Figure 4.12 (page 214). Find the following probabilities:

4.105 Uniform numbers between 0 and 2.

Many random number generators allow users to specify the range of the random numbers to be produced. Suppose that you specify that the range is to be all numbers between 0 and 2. Call the random number generated Y. Then the density curve of the random variable has constant height between 0 and 2, and height 0 elsewhere.

4.106 The sum of two uniform random numbers.

Generate two random numbers between 0 and 1 and take to be their sum. Then is a continuous random variable that can take any value between 0 and 2. The density curve of is the triangle shown in Figure 4.15.

4.107 How many close friends?

How many close friends do you have? Suppose that the number of close friends adults claim to have varies from person to person with mean and standard deviation . An opinion poll asks this question of an SRS of 1100 adults. We see in Chapter 6 that, in this situation, the sample mean response has approximately the Normal distribution with mean 9 and standard deviation 0.0724. What is , the probability that the statistic estimates to within ±1?

4.108 Normal approximation for a sample proportion.

A sample survey contacted an SRS of 700 registered voters in Oregon shortly after an election and asked respondents whether they had voted. Voter records show that 56% of registered voters had actually voted. We see in the next chapter that in this situation the proportion of the sample who voted has approximately the Normal distribution with mean and standard deviation .