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if X and Y are independent random variables (independence is not needed for the first of these three identities; for the second it can be wéakened to uncorrelatedness).

The central moments beyond the third lack this linéarity; in that respect they differ from the cumulants (the first three cumulants are the same as the first moment and the second and third central moments; the higher cumulants have a more complicated relationship with the central moments).

Like the cumulants, the factorial moments of a probability distribution are also polynomial functions of the moments.