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In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional normal distribution to higher dimensions.
In the event that the variables X and Y are jointly normally distributed random variables, then X + Y is still normally distributed (see Multivariate normal distribution) and the mean is the sum of the means. However, the variances are not additive due to the correlation. Indeed,
The multivariate normal distribution is a special case of the elliptical distributions. As such, its iso-density loci in the k = 2 case are ellipses and in the case of arbitrary k are ellipsoids. Rectified Gaussian distribution a rectified version of normal distribution with all the negative elements reset to 0
The multinomial distribution, a generalization of the binomial distribution. The multivariate normal distribution, a generalization of the normal distribution. The multivariate t-distribution, a generalization of the Student's t-distribution. The negative multinomial distribution, a generalization of the negative binomial distribution.
Multivariate normal distribution, which is the limiting case of the multivariate Student's t-distribution when . Chi distribution , the pdf of the scaling factor in the construction the Student's t-distribution and also the 2-norm (or Euclidean norm ) of a multivariate normally distributed vector (centered at zero).
The probability density function for the random matrix X (n × p) that follows the matrix normal distribution , (,,) has the form: (,,) = ([() ()]) / | | / | | /where denotes trace and M is n × p, U is n × n and V is p × p, and the density is understood as the probability density function with respect to the standard Lebesgue measure in , i.e.: the measure corresponding to integration ...
In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated normal distribution has wide applications in statistics and econometrics.
This measure is also known as the joint probability distribution, the joint distribution, or the multivariate distribution of the random vector. The distributions of each of the component random variables X i {\displaystyle X_{i}} are called marginal distributions .