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Bernoulli Random Variable
In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli,James Victor Uspensky: ''Introduction to Mathematical Probability'', McGraw-Hill, New York 1937, page 45 is the discrete probability distribution of a random variable which takes the value 1 with probability p and the value 0 with probability q = 1-p. Less formally, it can be thought of as a model for the set of possible outcomes of any single experiment that asks a yes–no question. Such questions lead to outcomes that are boolean-valued: a single bit whose value is success/yes/true/one with probability ''p'' and failure/no/ false/zero with probability ''q''. It can be used to represent a (possibly biased) coin toss where 1 and 0 would represent "heads" and "tails", respectively, and ''p'' would be the probability of the coin landing on heads (or vice versa where 1 would represent tails and ''p'' would be the probability of tails). In particular, unfair coins wou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Distribution
In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli,James Victor Uspensky: ''Introduction to Mathematical Probability'', McGraw-Hill, New York 1937, page 45 is the discrete probability distribution of a random variable which takes the value 1 with probability p and the value 0 with probability q = 1-p. Less formally, it can be thought of as a model for the set of possible outcomes of any single experiment that asks a yes–no question. Such questions lead to outcomes that are boolean-valued: a single bit whose value is success/yes/ true/one with probability ''p'' and failure/no/ false/ zero with probability ''q''. It can be used to represent a (possibly biased) coin toss where 1 and 0 would represent "heads" and "tails", respectively, and ''p'' would be the probability of the coin landing on heads (or vice versa where 1 would represent tails and ''p'' would be the probability of tails). In particular, unfair c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Mass Function
In probability and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete density function. The probability mass function is often the primary means of defining a discrete probability distribution, and such functions exist for either scalar or multivariate random variables whose domain is discrete. A probability mass function differs from a probability density function (PDF) in that the latter is associated with continuous rather than discrete random variables. A PDF must be integrated over an interval to yield a probability. The value of the random variable having the largest probability mass is called the mode. Formal definition Probability mass function is the probability distribution of a discrete random variable, and provides the possible values and their associated probabilities. It is the function p: \R \to ,1/math> defined by for -\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Categorical Distribution
In probability theory and statistics, a categorical distribution (also called a generalized Bernoulli distribution, multinoulli distribution) is a discrete probability distribution that describes the possible results of a random variable that can take on one of ''K'' possible categories, with the probability of each category separately specified. There is no innate underlying ordering of these outcomes, but numerical labels are often attached for convenience in describing the distribution, (e.g. 1 to ''K''). The ''K''-dimensional categorical distribution is the most general distribution over a ''K''-way event; any other discrete distribution over a size-''K'' sample space is a special case. The parameters specifying the probabilities of each possible outcome are constrained only by the fact that each must be in the range 0 to 1, and all must sum to 1. The categorical distribution is the generalization of the Bernoulli distribution for a categorical random variable, i.e. for a dis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sum Of Independent Random Variables
In probability theory and statistics, there are several relationships among probability distributions. These relations can be categorized in the following groups: *One distribution is a special case of another with a broader parameter space *Transforms (function of a random variable); *Combinations (function of several variables); *Approximation (limit) relationships; *Compound relationships (useful for Bayesian inference); * Duality; *Conjugate priors. Special case of distribution parametrization * A binomial distribution with parameters ''n'' = 1 and ''p'' is a Bernoulli distribution with parameter ''p''. * A negative binomial distribution with parameters ''n'' = 1 and ''p'' is a geometric distribution with parameter ''p''. * A gamma distribution with shape parameter ''α'' = 1 and rate parameter ''β'' is an exponential distribution with rate parameter ''β''. * A gamma distribution with shape parameter ''α'' = ''v''/2 and rate parameter ''β'' = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
