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According to this definition, E[X] exists and is finite if and only if E[X +] and E[X −] are both finite. Due to the formula |X| = X + + X −, this is the case if and only if E|X| is finite, and this is equivalent to the absolute convergence conditions in the definitions above. As such, the present considerations do not define finite ...
Given a discrete random variable X of ordered values to be encoded, let () be the probability for any x in X.Define a function ¯ = < + Algorithm: For each x in X, Let Z be the binary expansion of ¯ ().
In probability theory and statistics, the factorial moment generating function (FMGF) of the probability distribution of a real-valued random variable X is defined as = [] for all complex numbers t for which this expected value exists.
The probability generating function of an almost surely constant random variable, i.e. one with (=) = and () = is G ( z ) = z c . {\displaystyle G(z)=z^{c}.} The probability generating function of a binomial random variable , the number of successes in n {\displaystyle n} trials, with probability p {\displaystyle p} of success in each trial, is
In mathematics, the second moment method is a technique used in probability theory and analysis to show that a random variable has positive probability of being positive. More generally, the "moment method" consists of bounding the probability that a random variable fluctuates far from its mean, by using its moments. [1]
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).
For example, if the population is represented by bit strings of length 4, the EDA can represent the population of promising solution using a single vector of four probabilities (p1, p2, p3, p4) where each component of p defines the probability of that position being a 1. Using this probability vector it is possible to create an arbitrary number ...
In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate; the distance parameter could be any meaningful mono-dimensional measure of the process, such as time ...