Probability theory is the branch of mathematics concerned with probability. Sampling theory is part of the mathematical discipline of probability theory. Risk is defined as the product of probability and consequences. It defines an important concept of standard deviation used in probability theory and statistics. It allowed a decision to be made without the calculation of a probability. There are many alternatives for estimating the probability of default. It is equivalent to the following elementary result from probability theory. The book describes methods of mathematical model-building using probability and statistics. These interactions led to the conception of basic probability theory. They are also often useful in computations in elementary probability theory. See Cumulants of some discrete probability distributions. There are several ways of deriving formulae for the convolution of probability distributions. Subexponentiality is defined in terms of convolutions of probability distributions. This is the definition used in measure theory and probability. This leads to small values having a much higher probability than large values.