In probability theory, when is it appropriate to calculate the $\mathrm{expected value}$ of a random variable?

• A. When you need to find the probability of a single specific outcome occurring
• B. When you are interested in the maximum possible value a random variable can take
• C. When you want to calculate the $\mathrm{standard deviation}$ of a data set
• D. When you want to determine the long-run average outcome of a random process after many repetitions

Answer: (D)

Verified step by step guidance

1. Understand that the expected value (or expectation) of a random variable represents the long-run average outcome if the random process is repeated many times under the same conditions.
2. Recognize that the expected value is not used to find the probability of a single specific outcome; probabilities of specific outcomes are calculated differently, often using the probability mass or density function.
3. Note that the expected value does not directly give the maximum possible value of a random variable; it is a measure of central tendency, not an extreme value.
4. Understand that the expected value is different from the standard deviation, which measures the spread or variability of the data, not the average outcome.
5. Therefore, it is appropriate to calculate the expected value when you want to determine the long-run average outcome of a random process after many repetitions.