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When "E" is used to denote "expected value", authors use a variety of stylizations: the expectation operator can be stylized as E (upright), E (italic), or (in blackboard bold), while a variety of bracket notations (such as E(X), E[X], and EX) are all used. Another popular notation is μ X.
In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate; the distance parameter could be any meaningful mono-dimensional measure of the process, such as time ...
for -measurable , we have ((())) =, i.e. the conditional expectation () is in the sense of the L 2 (P) scalar product the orthogonal projection from to the linear subspace of -measurable functions. (This allows to define and prove the existence of the conditional expectation based on the Hilbert projection theorem .)
A different technique, which goes back to Laplace (1812), [3] is the following. Let = =. Since the limits on s as y → ±∞ depend on the sign of x, it simplifies the calculation to use the fact that e −x 2 is an even function, and, therefore, the integral over all real numbers is just twice the integral from zero to infinity.
In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace.It is also sometimes called the double exponential distribution, because it can be thought of as two exponential distributions (with an additional location parameter) spliced together along the abscissa, although the term is also sometimes used to refer to ...
Energy statistics are functions of distances between statistical observations. Energy distance and E-statistic were considered as N -distances and N-statistic in Zinger A.A., Kakosyan A.V., Klebanov L.B. Characterization of distributions by means of mean values of some statistics in connection with some probability metrics, Stability Problems ...
In probability and statistics, an exponential family is a parametric set of probability distributions of a certain form, specified below. This special form is chosen for mathematical convenience, including the enabling of the user to calculate expectations, covariances using differentiation based on some useful algebraic properties, as well as for generality, as exponential families are in a ...
In the mathematical theory of probability, the expectiles of a probability distribution are related to the expected value of the distribution in a way analogous to that in which the quantiles of the distribution are related to the median.