Role in Free Probability Theory
The lattice of noncrossing partitions plays the same role in defining "free cumulants" in free probability theory that is played by the lattice of all partitions in defining joint cumulants in classical probability theory. To be more precise, let be a non-commutative probability space, a non-commutative random variable with free cumulants . (See free probability for terminology.) Then
where denotes the number of blocks of length in the non-crossing partition . That is, the moments of a non-commutative random variable can be expressed as a sum of free cumulants over the sum non-crossing partitions. This is the free analogue of the moment-cumulant formula in classical probability. See also Wigner semicircle distribution.
Read more about this topic: Noncrossing Partition
Famous quotes containing the words role in, role, free, probability and/or theory:
“My role in society, or any artist or poet’s role, is to try and express what we all feel. Not to tell people how to feel. Not as a preacher, not as a leader, but as a reflection of us all.”
—John Lennon (1940–1980)
“A few [women] warrant our attention not because they have the answer but because they have rejected the mentality that insists there must be one answer. What makes them role models is not how much or how little they work, how many or how few hats they wear, but rather how well they understand, and accept, that for all rewards there will be commensurate sacrifice; for all gains, some loss; for any pleasure, some pain.”
—Melinda M. Marshall (20th century)
“For all the injustices in our past and our present, we have to believe that in the free exchange of ideas, justice will prevail over injustice, tolerance over intolerance and progress over reaction.”
—Hillary Rodham Clinton (b. 1947)
“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)
“[Anarchism] is the philosophy of the sovereignty of the individual. It is the theory of social harmony. It is the great, surging, living truth that is reconstructing the world, and that will usher in the Dawn.”
—Emma Goldman (1869–1940)