In
probability theory
Probability theory or probability calculus 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 expre ...
and
statistics
Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
, a stochastic order quantifies the concept of one
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
being "bigger" than another. These are usually
partial order
In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements needs to be comparable ...
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 rational choice is a branch of probability theory, probability, economics, and analytic philosophy that uses expected utility and probabilities, probability to model how individuals would behave Rationality, ratio ...
, under this circumstance, 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
,