Mellin Transform - in Probability Theory

In Probability Theory

In probability theory Mellin transform is an essential tool in studying the distributions of products of random variables. If X is a random variable, and X+ = max{X,0} denotes its positive part, while X − = max{−X,0} is its negative part, then the Mellin transform of X is defined as

\mathcal{M}_X(s) = \int_0^\infty x^s dF_{X^+}(x) + \gamma\int_0^\infty x^s dF_{X^-}(x),

where γ is a formal indeterminate with γ2 = 1. This transform exists for all s in some complex strip D = {s: a ≤ Re(s) ≤ b}, where a ≤ 0 ≤ b.

The Mellin transform of a random variable X uniquely determines its distribution function FX. The importance of the Mellin transform in probability theory lies in the fact that if X and Y are two independent random variables, then the Mellin transform of their products is equal to the product of the Mellin transforms of X and Y:

\mathcal{M}_{XY}(s) = \mathcal{M}_X(s)\mathcal{M}_Y(s)

Read more about this topic:  Mellin Transform

Famous quotes containing the words probability and/or theory:

    “Legends of prediction are common throughout the whole Household of Man. Gods speak, spirits speak, computers speak. Oracular ambiguity or statistical probability provides loopholes, and discrepancies are expunged by Faith.”
    —Ursula K. Le Guin (b. 1929)

    “Everything to which we concede existence is a posit from the standpoint of a description of the theory-building process, and simultaneously real from the standpoint of the theory that is being built. Nor let us look down on the standpoint of the theory as make-believe; for we can never do better than occupy the standpoint of some theory or other, the best we can muster at the time.”
    —Willard Van Orman Quine (b. 1908)