preserve non-negativity or total integral 1—but they are closed under convex combination, thus forming a convex subset of the space of functions (or measures). Equivalence of random variables There are several different senses in which random variables can be considered to be equivalent. Two random variables can be equal, equal almost surely, or equal in distribution. In increasing order of strength, the precise definition of these notions of equivalence is given below. Equality in distribution If the sample space is a subset of the real line, random variables X and Y are equal in distribution (denoted ) if they have the same distribution functions: To be equal in distribution, random variables need not be defined on the same probability space. Two random variables having equal moment generating functions have the same distribution. This provides, for example, a useful method of checking equality of certain functions of independent, identically distributed (IID) random variables. However, the moment generating function exists only for distributions that have a defined Laplace transform. Almost sure equality Two random variables X and Y are equal almost surely (denoted ) if, and only if, the probability that they are different is zero: For all practical purposes in probability theory, this notion of equivalence is as strong as actual equality. It is associated to the following distance: where "ess sup" represents the essential supremum in the sense of measure theory. Equality Finally, the two random variables X and Y are equal if they are equal as functions on their measurable space: This notion is typically the least useful in probability theory because in practice and in theory, the underlying measure space of the experiment is rarely explicitly characterized or even characterizable. Convergence A significant theme in mathematical statistics consists of obtaining convergence results for certain sequences of random variables; for instance the law of large numbers and the central limit theorem. There are various senses in which a sequence of random variables can converge to a random variable . These are explained in the article on convergence of random variables. See also Aleatoricism Algebra of random variables Event (probability theory) Multivariate random variable Pairwise independent random variables Observable variable Random compact set Random element Random function Random measure Random number generator produces a random value Random variate Random vector Randomness Stochastic process Relationships among probability distributions References Inline citations Literature External links Statistical randomness