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In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable.The general form of its probability density function is [2] [3] = ().
Gaussian functions are often used to represent the probability density function of a normally distributed random variable with expected value μ = b and variance σ 2 = c 2. In this case, the Gaussian is of the form [1]
[1] [2] In other words, () is the probability that a normal (Gaussian) random variable will obtain a value larger than standard deviations. Equivalently, () is the probability that a standard normal random variable takes a value larger than .
That is, for any two random variables X 1, X 2, both have the same probability distribution if and only if =. [ citation needed ] If a random variable X has moments up to k -th order, then the characteristic function φ X is k times continuously differentiable on the entire real line.
To obtain the marginal distribution over a subset of multivariate normal random variables, one only needs to drop the irrelevant variables (the variables that one wants to marginalize out) from the mean vector and the covariance matrix. The proof for this follows from the definitions of multivariate normal distributions and linear algebra.
A more general case of this concerns the distribution of the product of a random variable having a beta distribution with a random variable having a gamma distribution: for some cases where the parameters of the two component distributions are related in a certain way, the result is again a gamma distribution but with a changed shape parameter.
1.3 Example. 2 Proof. 3 ... 3.2 Non-Gaussian random variables. 3.3 Uniform ... An equivalent formulation of the Wick's probability formula is the Gaussian ...
An example found by Marcus and Shepp [18]: 387 is a random lacunary Fourier series = = ( + ), where ,,,, … are independent random variables with standard normal distribution; frequencies < < < … are a fast growing sequence; and coefficients > satisfy <.