Difference between revisions of "Random variable"
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Revision as of 07:16, 4 March 2013
2020 Mathematics Subject Classification: Primary: 60-01 [MSN][ZBL]
One of the basic concepts in probability theory. The role of random variables and their expectations (cf. Mathematical expectation) was clearly pointed out by P.L. Chebyshev (1867; see [C]). The realization that the concept of a random variable is a special case of the general concept of a measurable function came much later. A full exposition, free from any superfluous restrictions, of the basics of probability theory in a measure-theoretical setting was first given by A.N. Kolmogorov (1933; see [Ko]). This made it clear that a random variable is nothing but a measurable function on a probability space. This has to be clearly stated, even in an elementary exposition on probability theory. In the academic literature this point of view was adopted by W. Feller (see the foreword to [F], where the exposition is based on the concept of a space of elementary events, and where it is stressed that only in this context the notion of a random variable becomes meaningful).
Let 
be a probability space. A single-valued real-valued function 
defined on 
is called a random variable if for any real 
the set 
belongs to the class 
. Let 
be any random variable and 
the class of subsets 
for which 
; this is a 
-algebra. The class 
of all Borel subsets of 
is always contained in 
. The measure 
defined on 
by the equation 
, 
, is called the probability distribution of 
. This measure is uniquely determined by the distribution function of 
:
 |
The values 
for 
(that is, the values of a measure extending 
to 
) are not, in general, uniquely determined by 
(a sufficient condition for uniqueness is so-called perfectness of the measure 
; see Perfect measure, and also [GK]). This must constantly be borne in mind (for example, when proving that the distribution of a random variable is uniquely determined by its characteristic function).
If a random variable 
takes a finite or countable number of pairwise distinct values 
with probabilities 
(
), then its probability distribution (which is said to be discrete in this case) is given by
 |
The distribution of 
is called continuous if there is a function 
(called the probability density) such that
 |
for every interval 
(or equivalently, for every Borel set 
). In the usual terminology of mathematical analysis this means that 
is absolutely continuous with respect to Lebesgue measure on 
.
Several general properties of the probability distribution of a random variable 
are sufficiently characterized by a small number of numerical characteristics. For example, the median (in statistics) and quantiles (cf. Quantile) have the advantage that they are defined for all distributions, although the most widely used are the mathematical expectation 
and the dispersion (or variance) 
of 
. See also Probability theory.
A complex random variable 
is determined by a pair of real random variables 
and 
by the formula
 |
An ordered set 
of random variables can be regarded as a random vector with values in 
.
The notion of a random variable can be generalized to the infinite-dimensional case using the concept of a random element.
It is worth noting that in certain problems of mathematical analysis and number theory it is convenient to regard the functions involved in their formulation as random variables defined on suitable probability spaces (see [Ka] for example).
References
| [C] | P.L. Chebyshev, "On mean values" , Complete collected works , 2 , Moscow-Leningrad (1947) (In Russian) |
| [Ko] | A.N. Kolmogorov, "Grundbegriffe der Wahrscheinlichkeitsrechnung" , Springer (1973) (Translated from Russian) MR0362415 |
| [F] | W. Feller, "An introduction to probability theory and its applications", 1 , Wiley (1950) |
| [GK] | B.V. Gnedenko, A.N. Kolmogorov, "Limit distributions for sums of independent random variables" , Springer (1958) (Translated from Russian) MR0233400 MR0062975 MR0041377 Zbl 0056.36001 |
| [Ka] | M. Kac, "Statistical independence in probability, analysis and number theory" , Math. Assoc. Amer. (1959) MR0110114 Zbl 0088.10303 |
Comments
Other mathematicians adopting Kolmogorov's point of view are, e.g., J.L. Doob [D] and P. Lévy [L].
References
| [D] | J.L. Doob, "Stochastic processes depending on a continuous parameter" Trans. Amer. Math. Soc. , 42 (1937) pp. 107–140 MR1501916 Zbl 0017.02701 Zbl 63.1075.01 |
| [L] | P. Lévy, "Le mouvement brownien plan" Amer. J. Math. , 62 (1940) pp. 487–550 MR0002734 Zbl 0024.13906 Zbl 66.0619.02 |
Random variable. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Random_variable&oldid=29510