# Abstract Probability Distributions From an abstract mathematical perspective, probability is surprisingly boring: > **Probability**: Positive, conserved quantity that we want to *distribute* across a given space. Note that it does not refer to anything inherently random or uncertain. A **probability distribution** defines a mathematically self-consistent allocation of this conserved quantity across a space $X$. Put another way: > Formal probability theory is simply the study of probability distributions that distribute a finite, conserved quantity across a space, the expectation values that such a distribution induces, and how the distribution behaves under transformations of the underlying space. One of the challenges with probability distributions is that *most probability distributions cannot be explicitly defined in most problems*! Remember: A probability distribution is defined as a map from the well-defined subsets in a $\sigma$-algebra to real numbers in the interval. For a $\sigma$-algebra with only a few elements we could completely specify this mapping with a lookup table that associates a probability with each set. But in practice, almost all $\sigma$-algebras are simply too large for this to be practical. In order to implement probability distributions in practice we need a way to define this mapping algorithmically so that we can calculate probabilities and expectation values on the fly. Fortunately most probabilistic systems admit ***representations*** that faithfully recover all probabilities and expectation values on demand and hence completely specify a given probability distribution. In particular, density representations exploit the particular structure of X to fully characterize a probability distribution with special functions. --- Date: 20220522 Links to: Tags: #review References: * Notability, Probability Theory * [Probability Theory (For Scientists and Engineers)](https://betanalpha.github.io/assets/case_studies/probability_theory.html)