Yes, let's address each part of your question:

1. Can a discrete random variable have an infinite number of possible outcomes?

   • Yes, a discrete random variable can have an infinite number of possible outcomes. A key characteristic of a discrete random variable is that the outcomes can be listed or counted, even if the set is infinite. For example, the set of non-negative integers $\{0, 1, 2, 3, \dots\}$ is infinite, but each individual value can be counted. An example is the number of tosses until the first success in a geometric distribution.

2. Can a continuous random variable have a finite number of possible outcomes?

   • No, a continuous random variable cannot have a finite number of possible outcomes. By definition, a continuous random variable can take any value within an interval or collection of intervals on the real number line. The number of possible outcomes is uncountably infinite. Examples include the exact time it takes for a chemical reaction to complete or the height of people in a population.

Do you want more details or further clarifications?

Here are some related questions you may explore:

1. What are the key differences between discrete and continuous random variables?
2. Can a discrete random variable take non-integer values?
3. How are probability mass functions (PMFs) used for discrete random variables?
4. What is the concept of a probability density function (PDF) for continuous random variables?
5. Can a continuous variable ever have a probability of exactly zero for any single outcome?

Tip: For continuous random variables, the probability of taking any specific value is always zero, since probabilities are associated with intervals rather than specific points.