takes on a value in a measurable set is written as Standard case In many cases, is real-valued, i.e. . In some contexts, the term random element (see extensions) is used to denote a random variable not of this form. When the image (or range) of is countable, the random variable is called a discrete random variable and its distribution is a discrete probability distribution, i.e. can be described by a probability mass function that assigns a probability to each value in the image of . If the image is uncountably infinite (usually an interval) then is called a continuous random variable. In the special case that it is absolutely continuous, its distribution can be described by a probability density function, which assigns probabilities to intervals; in particular, each individual point must necessarily have probability zero for an absolutely continuous random variable. Not all continuous random variables are absolutely continuous, a mixture distribution is one such counterexample; such random variables cannot be described by a probability density or a probability mass function. Any random variable can be described by its cumulative distribution function, which describes the probability that the random variable will be less than or equal to a certain value. Extensions The term "random variable" in statistics is traditionally limited to the real-valued case (). In this case, the structure of the real numbers makes it possible to define quantities such as the expected value and variance of a random variable, its cumulative distribution function, and the moments of its distribution. However, the definition above is valid for any measurable space of values. Thus one can consider random elements of other sets , such as random boolean values, categorical values, complex numbers, vectors, matrices, sequences, trees, sets, shapes, manifolds, and functions. One may then specifically refer to a random variable of type , or an -valued random variable. This more general concept of a random element is particularly useful in disciplines such as graph theory, machine learning, natural language processing, and other fields in discrete mathematics and computer science, where one is often interested in modeling the random variation of non-numerical data structures. In some cases, it is nonetheless convenient to represent each element of , using one or more real numbers. In this case, a random element may optionally be represented as a vector of real-valued random variables (all defined on the same underlying probability space , which