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Random Variables and their Characteristics

The reader may recall the well-known normal (Gaussian) distribution of a random variable. As a simple example, let random vector variable X take its values (x, ) in two-dimensional space, and let us consider the (so-called) [Pg.589]

Let US generalize. In terms rather intuitive than precise mathematically, a random variable X is an element of a given (say, iV-dimensional) vector space U, which can take different values in the manner that, whatever be a region (c ), [Pg.590]

Hence the probability cannot be negative, and the probability that the value of X is found somewhere in U equals unity. Moreover, let us adopt the condition that the function /x is (not only integrable, but also) sufficiently small at infinity see (E.l.lOa) below. [On the other hand, we do not require/x to be continuous.] In this manner, the randomness of variable X is quantitatively characterized by the joint probability density f. The probability is distributed according to the integrals (E.1.2). The law (E.1.2) is also called the probability distribution of random variable X The probability (of the event) that the value of X is found in 2 is determined by a well-defined integral over 2 . For a random vector variable (with N ), the distribution is called multivariate. [Pg.590]

It is rather a philosophical (onto- and epistemological) problem (and a very deep one), what is the probability. For a mathematician, the probabilistic concepts are introduced axiomatically and the laws are consequences of the axioms.So does also theoretical statistics, on introducing further concepts idealizing the (physical, industrial, economic, - ) reality. [In practice, the abstract theory is tacitly forgotten and replaced by routine formulae.] In this book, we can manage with a limited number of theoretical concepts. The basic ones are the following. [Pg.590]

Generally, if g is a function defined on P, let g(X) be the (random) variable associating the value g(x) with any value x of X in vector space Then the integral mean value of g, with density /x, equals by definition [Pg.590]


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