allows the different random variables to covary). For example: A random word may be represented as a random integer that serves as an index into the vocabulary of possible words. Alternatively, it can be represented as a random indicator vector, whose length equals the size of the vocabulary, where the only values of positive probability are , , and the position of the 1 indicates the word. A random sentence of given length may be represented as a vector of random words. A random graph on given vertices may be represented as a matrix of random variables, whose values specify the adjacency matrix of the random graph. A random function may be represented as a collection of random variables , giving the function's values at the various points in the function's domain. The are ordinary real-valued random variables provided that the function is real-valued. For example, a stochastic process is a random function of time, a random vector is a random function of some index set such as , and random field is a random function on any set (typically time, space, or a discrete set). Distribution functions If a random variable defined on the probability space is given, we can ask questions like "How likely is it that the value of is equal to 2?". This is the same as the probability of the event which is often written as or for short. Recording all these probabilities of outputs of a random variable yields the probability distribution of . The probability distribution "forgets" about the particular probability space used to define and only records the probabilities of various output values of . Such a probability distribution, if is real-valued, can always be captured by its cumulative distribution function and sometimes also using a probability density function, . In measure-theoretic terms, we use the random variable to "push-forward" the measure on to a measure on . The measure is called the "(probability) distribution of " or the "law of ". The density , the Radon–Nikodym derivative of with respect to some reference measure on (often, this reference measure is the Lebesgue measure in the case of continuous random variables, or the counting measure in the case of discrete random variables). The underlying probability space is a technical device used to guarantee the existence of random variables, sometimes to construct them, and to define notions such as correlation and dependence or