DISCRETE INTERPOLATION BETWEEN MONOTONE PROBABILITY AND FREE PROBABILITY

Probability and Interpolation

An m x n matrix E with n ones and (m — l)/i zeros, which satisfies the Polya condition, may be regular and singular for Birkhoff interpolation. We prove that for random distributed ones, E is singular with probability that converges to one if m, n -» oo. Previously, this was known only if m > (1 + 8)n/log n. For constant m and n -» oo, the probability is asymptotically at least i.

Discrete Mathematics and Probability

Discrete Probability Refresher

Probability theory plays a central role in information theory. Information sources and communications channels are modelled probabilistically, and the key measures of information theory (like entropy and channel capacity) are defined in terms of the underlying random variables. The student of information theory is expected to have some familiarity with probability and the theory of random varia...

Discrete Mathematics and Probability Theory

Combinatorial Mathematics and Discrete Probability

Consider the experiment of tossing a 6-sided die. Suppose the outcome is 2. Does the event that the outcome is even happen? The answer is yes. Now assume that the toss produces an outcome 4, does the event that the outcome is even happen? The answer is yes. Actually observe that when the outcome is from the set {2, 4, 6}, the event happens. Thus an event in a probabilistic experiment is actuall...