Linearity of expectation states that for any random variables X and Y (independent or not, discrete or continuous), E(X+Y) = E(X) + E(Y), and more generally E(cX) = cE(X) for a constant c; the proof reframes expectation as a sum over sample-space outcomes ("pebbles") rather than over the random variable's values, so that summation trivially distributes across a sum of functions. This principle belongs to probability theory's treatment of expectation and is a foundational tool used alongside indicator random variables to compute expectations of complex counting quantities without needing independence or joint distributions.
Linearity of expectation states that for any random variables X and Y (independent or not, discrete or continuous), E(X+Y) = E(X) + E(Y), and more generally E(cX) = cE(X) for a constant c; the proof …