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Product Distribution
A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables ''X'' and ''Y'', the distribution of the random variable ''Z'' that is formed as the product Z = XY is a ''product distribution''. Algebra of random variables The product is one type of algebra for random variables: Related to the product distribution are the ratio distribution, sum distribution (see List of convolutions of probability distributions) and difference distribution. More generally, one may talk of combinations of sums, differences, products and ratios. Many of these distributions are described in Melvin D. Springer's book from 1979 ''The Algebra of Random Variables''. Derivation for independent random variables If X and Y are two independent, continuous random variables, described by probability density functions f_X and f_Y then the probability density ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space). For instance, if is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of would take the value 0.5 (1 in 2 or 1/2) for , and 0.5 for (assuming that the coin is fair). Examples of random phenomena include the weather conditions at some future date, the height of a randomly selected person, the fraction of male students in a school, the results of a survey to be conducted, etc. Introduction A probability distribution is a mathematical description of the probabilities of events, subsets of the sample space. The sample space, often denoted by \Omega, is the set of all possible outcomes of a random phe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gamma Distribution
In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma distribution. There are two equivalent parameterizations in common use: #With a shape parameter k and a scale parameter \theta. #With a shape parameter \alpha = k and an inverse scale parameter \beta = 1/ \theta , called a rate parameter. In each of these forms, both parameters are positive real numbers. The gamma distribution is the maximum entropy probability distribution (both with respect to a uniform base measure and a 1/x base measure) for a random variable X for which E 'X''= ''kθ'' = ''α''/''β'' is fixed and greater than zero, and E n(''X'')= ''ψ''(''k'') + ln(''θ'') = ''ψ''(''α'') − ln(''β'') is fixed (''ψ'' is the digamma function). Definitions The parameterization with ''k'' and ''θ'' appears to be more common in econo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wishart Distribution
In statistics, the Wishart distribution is a generalization to multiple dimensions of the gamma distribution. It is named in honor of John Wishart, who first formulated the distribution in 1928. It is a family of probability distributions defined over symmetric, nonnegative-definite random matrices (i.e. matrix-valued random variables). In random matrix theory, the space of Wishart matrices is called the ''Wishart ensemble''. These distributions are of great importance in the estimation of covariance matrices in multivariate statistics. In Bayesian statistics, the Wishart distribution is the conjugate prior of the inverse covariance-matrix of a multivariate-normal random-vector. Definition Suppose is a matrix, each column of which is independently drawn from a -variate normal distribution with zero mean: :G_ = (g_i^1,\dots,g_i^p)^T\sim \mathcal_p(0,V). Then the Wishart distribution is the probability distribution of the random matrix :S= G G^T = \sum_^n G_G_^T kno ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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