Linearity of expectation states that the expected value of a sum of random variables equals the sum of their individual expected values, regardless of whether the variables are independent; this permits decomposing a complicated random variable into a sum of simpler, tractable components (often geometric waiting times) to compute its expectation directly. The coupon collector problem — the expected number of draws needed to obtain a complete set of n equally likely item types — is a canonical application: the total collection time is decomposed into successive "waiting for the next new type" stages, each geometrically distributed, yielding an expectation equal to n times the nth harmonic number. This belongs to probability theory, specifically the theory of expectation and its structural properties (linearity, symmetry, the law of the unconscious statistician), which generalizes across discrete and continuous distributions.
Linearity of expectation states that the expected value of a sum of random variables equals the sum of their individual expected values, regardless of whether the variables are independent; this perm…