. . . . . . . . random variables are used to model rates.

The correct answer is D. All of the mentioned.

Empirical random variables are used to model rates when the data is collected from a real-world experiment. Binomial random variables are used to model rates when the number of trials is fixed and the probability of success is constant. Poisson random variables are used to model rates when the number of trials is large and the probability of success is small.

Here is a more detailed explanation of each option:

• Empirical random variables are used to model rates when the data is collected from a real-world experiment. For example, if we want to model the rate of car accidents in a city, we could collect data on the number of car accidents that occur each month for a period of several years. We could then use this data to estimate the probability of a car accident occurring in any given month.
• Binomial random variables are used to model rates when the number of trials is fixed and the probability of success is constant. For example, if we want to model the rate of success of a new drug, we could conduct a clinical trial in which we give the drug to a group of patients and see how many of them experience a positive outcome. We could then use this data to estimate the probability of a patient experiencing a positive outcome if they take the drug.
• Poisson random variables are used to model rates when the number of trials is large and the probability of success is small. For example, if we want to model the rate of phone calls that come into a call center, we could collect data on the number of phone calls that come in each hour for a period of several weeks. We could then use this data to estimate the average number of phone calls that come in each hour.

In conclusion, all of the mentioned random variables are used to model rates. The choice of which random variable to use depends on the specific situation.