How is the expected value of a random variable calculated?

• A. As the average of the random variable outcomes
• B. As the sum of all possible values without weighting
• C. As the sum of all possible values weighted by their probabilities
• D. As the product of probabilities and values

Answer: (C)

The expected value of a random variable is calculated by taking the sum of all possible values of the variable, each multiplied by its probability of occurring. This approach reflects the average outcome one would anticipate from an experiment in the long run, providing a weighted average that accounts for the likelihood of each outcome.

In practical terms, if you have a random variable that can take on different values, the expected value aggregates these values in a way that considers how often each one is expected to occur. For instance, if you roll a fair six-sided die, the expected value can be calculated as follows: you take each outcome (1, 2, 3, 4, 5, and 6), multiply it by the probability of rolling that number (which is 1/6 for each number), and sum those products together. This provides a concise and accurate measure of the center of the distribution of outcomes.

The other options do not correctly describe the method for calculating expected value. Simply averaging outcomes without weighting by probabilities would not yield an accurate representation, as it ignores the frequency of each outcome. Summing all possible values without weighting does not provide a measure of expectation that takes into account how likely each outcome is. Similarly, stating it as the product