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Any definition of expected value may be extended to define an expected value of a multidimensional random variable, i.e. a random vector X. It is defined component by component, as E[X] i = E[X i]. Similarly, one may define the expected value of a random matrix X with components X ij by E[X] ij = E[X ij].
The Cauchy distribution is often used in statistics as the canonical example of a "pathological" distribution since both its expected value and its variance are undefined (but see § Moments below). The Cauchy distribution does not have finite moments of order greater than or equal to one; only fractional absolute moments exist. [1]
For some probability distributions, the expected value may be infinite or undefined, but if defined, it is unique. The mean of a (finite) sample is always defined. The median is the value such that the fractions not exceeding it and not falling below it are each at least 1/2. It is not necessarily unique, but never infinite or totally undefined.
In general, uncorrelatedness is not the same as orthogonality, except in the special case where at least one of the two random variables has an expected value of 0. In this case, the covariance is the expectation of the product, and X {\displaystyle X} and Y {\displaystyle Y} are uncorrelated if and only if E [ X Y ] = 0 {\displaystyle ...
In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] or (0, 1) in terms of two positive parameters, denoted by alpha (α) and beta (β), that appear as exponents of the variable and its complement to 1, respectively, and control the shape of the distribution.
In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value evaluated with respect to the conditional probability distribution. If the random variable can take on only a finite number of values, the "conditions" are that the variable can only take on a subset of ...
An undefined value must not be confused with empty string, Boolean "false" or other "empty" (but defined) values. Depending on circumstances, evaluation to an undefined value may lead to exception or undefined behaviour, but in some programming languages undefined values can occur during a normal, predictable course of program execution.
if >, else undefined: Median: Mode (+) / if >, 0 ... is the expected value. For the log-logistic distribution, the formula for the Gini coefficient becomes: ...