Expected value (equivalently, the mean or average) of a random variable is defined as the probability-weighted sum of its possible outcomes, and is distinct from the median, which is the value splitting the probability mass in half. Linearity of expectation states that the expected value of a sum of random variables equals the sum of their expected values, regardless of whether the variables are independent, making it one of the most powerful tools in probability theory for computing expectations of complex random variables by decomposing them into simpler indicator random variables. This body of theory belongs to discrete probability theory and underlies applications such as mean-time-to-failure analysis, expected-return analysis of wagering schemes, and combinatorial expectation problems (e.g., the hat-check / random-permutation problem).
Expected value (equivalently, the mean or average) of a random variable is defined as the probability-weighted sum of its possible outcomes, and is distinct from the median, which is the value splitt…