What is the formula to calculate the long-term average of a discrete random variable?

The formula E(X) = Σ [P(x) * x] is used to calculate the expected value, or long-term average, of a discrete random variable. In this context, P(x) represents the probability of the random variable taking on the value x, and the summation is taken over all possible values of x.

To understand why this formula works, consider that the expected value provides a weighted average of all possible outcomes, where the weights are the probabilities of those outcomes. By multiplying each outcome (x) by its associated probability (P(x)), we account for how likely each outcome is to occur. Adding up these products across all possible values gives us a comprehensive average that reflects both the values and their likelihoods.

This approach emphasizes that not all outcomes contribute equally to the average, as rarer events have a smaller cumulative impact compared to more probable outcomes. Therefore, using the formula effectively incorporates the probabilities, allowing us to derive a realistic expectation for what we can anticipate from the random variable over the long term.

Understanding this concept is essential in fields that involve statistical analysis, risk assessment, or decision-making based on uncertain outcomes.