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

• A. When you want to measure the $\mathrm{spread}$ or $\mathrm{variability}$ of a distribution
• B. When you want to determine the $\mathrm{long‑run average}$ outcome of a random process after many repetitions
• C. When you want to find the $\mathrm{maximum}$ possible value a random variable can take
• D. When you want to calculate the $\mathrm{probability}$ of a single specific outcome

Answer: (B)

Verified step by step guidance

1. Understand that the expected value (or mean) of a random variable represents the long-run average outcome if the random process is repeated many times.
2. Recognize that the expected value is not used to measure spread or variability; those are described by other measures like variance or standard deviation.
3. Note that the expected value is not about finding the maximum or minimum possible values of the random variable; it is a weighted average of all possible values.
4. Also, the expected value is different from calculating the probability of a single specific outcome, which is simply the chance that one particular event occurs.
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.