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Papoulis contributed in the areas of signal processing, communications, and signal and system theory. His classic book Probability, Random Variables, and Stochastic Processes [4] is used as a textbook in many graduate-level probability courses in electrical engineering departments all over the world.
In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no question, and each with its own Boolean-valued outcome: success (with probability p) or failure (with probability q = 1 − p).
The Optimum "L" filter (also known as a Legendre–Papoulis filter) was proposed by Athanasios Papoulis in 1958. It has the maximum roll off rate for a given filter order while maintaining a monotonic frequency response .
The probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function or probability density function of a sum of independent random variables is the convolution of their corresponding probability mass functions or probability density functions respectively.
A stochastic process is defined as a collection of random variables defined on a common probability space (,,), where is a sample space, is a -algebra, and is a probability measure; and the random variables, indexed by some set , all take values in the same mathematical space , which must be measurable with respect to some -algebra .
The Papoulis-Marks-Cheung approach [1] is a theorem in multidimensional Shannon sampling theory that shows that the sampling density of a two-dimensional bandlimited function can be reduced to the support of the Fourier transform of the function.
If, for example, there are two balls and three bins, then the number of ways of placing the balls is (+) = =. The table shows the six possible ways of distributing the two balls, the strings of stars and bars that represent them (with stars indicating balls and bars separating bins from one another), and the subsets that correspond to the strings.
For example, Jackson's theorem gives the joint equilibrium distribution of an open queueing network as the product of the equilibrium distributions of the individual queues. [1] After numerous extensions, chiefly the BCMP network it was thought local balance was a requirement for a product-form solution.