How is the expected value (E(X)) of a discrete random variable calculated?

The expected value ( E(X) ) of a discrete random variable is calculated using the formula ( E(X) = \Sigma [x * P(x)] ), where ( x ) represents the possible values of the random variable and ( P(x) ) represents the probability associated with each value ( x ).

This formula essentially weights each outcome ( x ) by its probability ( P(x) ) and sums these products across all possible outcomes. This gives a measure of the center of the distribution of the random variable, essentially providing a "long-term average" of the outcomes if the random experiment were repeated many times.

Understanding this calculation is pivotal as it helps in various applications, such as predicting outcomes in decision-making scenarios, evaluating risk, and optimizing strategies across diverse fields such as economics, finance, and statistics. The expected value effectively summarizes the overall tendency of the random variable, allowing for informed conclusions based on the calculated probabilities.