In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
and
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, a stochastic order quantifies the concept of one
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
being "bigger" than another. These are usually
partial order
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
s, so that one random variable
may be neither stochastically greater than, less than nor equal to another random variable
. Many different orders exist, which have different applications.
Usual stochastic order
A real random variable
is less than a random variable
in the "usual stochastic order" if
:
where
denotes the probability of an event. This is sometimes denoted
or
. If additionally
for some
, then
is stochastically strictly less than
, sometimes denoted
. In
decision theory
Decision theory (or the theory of choice; not to be confused with choice theory) is a branch of applied probability theory concerned with the theory of making decisions based on assigning probabilities to various factors and assigning numerical ...
, under this circumstance ''B'' is said to be
first-order stochastically dominant over ''A''.
Characterizations
The following rules describe situations when one random variable is stochastically less than or equal to another. Strict version of some of these rules also exist.
#
if and only if for all non-decreasing functions
,