Given a table that lists the possible values of a random variable along with their corresponding probabilities, is the random variable $X$ discrete or continuous?

• A. Discrete, because the random variable $X$ can take on only a countable number of distinct values.
• B. Continuous, because the probabilities are assigned to intervals rather than individual values.
• C. Discrete, because the random variable $X$ can take on any value within a range.
• D. Continuous, because the random variable $X$ can take on any value within an interval.

Answer: (A)

Verified step by step guidance

1. Step 1: Understand the definitions of discrete and continuous random variables. A discrete random variable takes on countable, distinct values, often integers or specific points, while a continuous random variable can take on any value within an interval or range, including infinitely many possible values.
2. Step 2: Examine the given table that lists possible values of the random variable along with their corresponding probabilities. If the probabilities are assigned to specific individual values, this suggests the variable is discrete.
3. Step 3: Recognize that for continuous random variables, probabilities are not assigned to individual points but rather to intervals, because the probability of any single exact value is zero in a continuous distribution.
4. Step 4: Based on the information that the table lists specific values and their probabilities, conclude that the random variable is discrete, as it takes on countable distinct values with assigned probabilities.
5. Step 5: Summarize that the correct reasoning is: the random variable is discrete because it can take on only a countable number of distinct values, each with an associated probability.