Linearity of expectation

Linearity of expectation states that the expected value of a finite sum of random variables equals the sum of their expected values, whether or not they are independent.

Expected value: A probability-weighted average of the possible values of a random variable. Linearity applies to expected values, which average each variable over its probability distribution.
Indicator method: A technique that represents a count as a sum of indicator random variables. It uses linearity to find expected counts without modeling dependence among events.
Independence (probability theory): A condition in which knowing one event or random variable does not change the probabilities of another. Linearity needs no independence, though independence can simplify other calculations involving sums.
Random variable: A function that assigns a numerical value to each outcome in a random experiment. The rule combines expectations of random variables, which need not be independent.
Indicator random variable: A random variable equal to one when an event occurs and zero otherwise. Expressing counts as sums of indicators makes their expectations easy to add.
Coupon collector problem: A probability problem about how many random draws are needed to collect every type of coupon. Its expected collection time is derived by summing waiting-time expectations for successive new types.
Variance: A measure of the spread of a random variable around its expected value. Unlike expectation, variance of a sum generally includes dependence through covariance terms.
Event (probability theory): A set of outcomes in a probability space. Indicators turn events into zero-one variables whose expectations equal event probabilities.
Expectation operator: The operation that assigns a random variable its expected value. Its additive property is the formal statement of this rule.
Expected number of inversions: The expected count of out-of-order pairs in a randomly ordered sequence. Each pair contributes an indicator, so the total expectation is a sum of pairwise probabilities.

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