any generating set. Here we can prove measurability on this generating set by using the fact that . Moments The probability distribution of a random variable is often characterised by a small number of parameters, which also have a practical interpretation. For example, it is often enough to know what its "average value" is. This is captured by the mathematical concept of expected value of a random variable, denoted , and also called the first moment. In general, is not equal to . Once the "average value" is known, one could then ask how far from this average value the values of typically are, a question that is answered by the variance and standard deviation of a random variable. can be viewed intuitively as an average obtained from an infinite population, the members of which are particular evaluations of . Mathematically, this is known as the (generalised) problem of moments: for a given class of random variables , find a collection of functions such that the expectation values fully characterise the distribution of the random variable . Moments can only be defined for real-valued functions of random variables (or complex-valued, etc.). If the random variable is itself real-valued, then moments of the variable itself can be taken, which are equivalent to moments of the identity function of the random variable. However, even for non-real-valued random variables, moments can be taken of real-valued functions of those variables. For example, for a categorical random variable X that can take on the nominal values "red", "blue" or "green", the real-valued function can be constructed; this uses the Iverson bracket, and has the value 1 if has the value "green", 0 otherwise. Then, the expected value and other moments of this function can be determined. Functions of random variables A new random variable Y can be defined by applying a real Borel measurable function to the outcomes of a real-valued random variable . That is, . The cumulative distribution function of is then If function is invertible (i.e., exists, where is 's inverse function) and is either increasing or decreasing, then the previous relation can be extended to obtain With the same hypotheses of invertibility of , assuming also differentiability, the relation between the probability density functions can be found by differentiating both sides of the above expression with respect to , in order to obtain If there is no invertibility of but each