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The Cauchy distribution, named after Augustin-Louis Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz (ian) function, or Breit–Wigner distribution.
The reciprocal 1/ X of a random variable X, is a member of the same family of distribution as X, in the following cases: Cauchy distribution, F distribution, log logistic distribution. Examples: If X is a Cauchy (μ, σ) random variable, then 1/ X is a Cauchy (μ / C, σ / C) random variable where C = μ2 + σ2. If X is an F (ν1, ν2) random ...
In probability theory, a log-Cauchy distribution is a probability distribution of a random variable whose logarithm is distributed in accordance with a Cauchy distribution. If X is a random variable with a Cauchy distribution, then Y = exp (X) has a log-Cauchy distribution; likewise, if Y has a log-Cauchy distribution, then X = log (Y) has a ...
The characteristic function is a way to describe a random variable. The characteristic function, a function of t, determines the behavior and properties of the probability distribution of the random variable X. It is equivalent to a probability density function or cumulative distribution function in the sense that knowing one of the functions ...
In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be stable if its distribution is stable. The stable distribution family is also sometimes referred to as the ...
In probability theory and statistics, an inverse distribution is the distribution of the reciprocal of a random variable. Inverse distributions arise in particular in the Bayesian context of prior distributions and posterior distributions for scale parameters. In the algebra of random variables, inverse distributions are special cases of the ...
Examples of continuous distributions that are infinitely divisible are the normal distribution, the Cauchy distribution, the Lévy distribution, and all other members of the stable distribution family, as well as the Gamma distribution, the chi-square distribution, the Wald distribution, the Log-normal distribution [2] and the Student's t-distribution.
In probability theory, the law of large numbers (LLN) is a mathematical law that states that the average of the results obtained from a large number of independent random samples converges to the true value, if it exists. [1] More formally, the LLN states that given a sample of independent and identically distributed values, the sample mean ...