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The distribution N(μ, Σ) is in effect N(0, I) scaled by Λ 1/2, rotated by U and translated by μ. Conversely, any choice of μ, full rank matrix U, and positive diagonal entries Λ i yields a non-singular multivariate normal distribution. If any Λ i is zero and U is square, the resulting covariance matrix UΛU T is singular.
The scaled sum of a sequence of i.i.d. random variables with finite positive variance converges in distribution to the normal distribution. In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the distribution of a normalized version of the sample mean converges to a standard normal distribution.
In statistics, an F-test of equality of variances is a test for the null hypothesis that two normal populations have the same variance.Notionally, any F-test can be regarded as a comparison of two variances, but the specific case being discussed in this article is that of two populations, where the test statistic used is the ratio of two sample variances. [1]
Conditional probability distribution. In probability theory and statistics, the conditional probability distribution is a probability distribution that describes the probability of an outcome given the occurrence of a particular event. Given two jointly distributed random variables and , the conditional probability distribution of given is the ...
considered as a function of , is the likelihood function, given the outcome of the random variable . Sometimes the probability of "the value of for the parameter value " is written as P(X = x | θ) or P(X = x; θ). The likelihood is the probability that a particular outcome is observed when the true value of the parameter is , equivalent to the ...
Normality test. In statistics, normality tests are used to determine if a data set is well-modeled by a normal distribution and to compute how likely it is for a random variable underlying the data set to be normally distributed. More precisely, the tests are a form of model selection, and can be interpreted several ways, depending on one's ...
The cumulative distribution function of a real-valued random variable is the function given by [2]: p. 77. (Eq.1) where the right-hand side represents the probability that the random variable takes on a value less than or equal to . The probability that lies in the semi-closed interval , where , is therefore [2]: p. 84.
About 68% of values drawn from a normal distribution are within one standard deviation σ from the mean; about 95% of the values lie within two standard deviations; and about 99.7% are within three standard deviations. [6] This fact is known as the 68–95–99.7 (empirical) rule, or the 3-sigma rule.