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# CHAPTER 15Univariate Statistical Distributions

## 15.1 Introduction, Goals and Objectives

In this chapter we give an introduction to univariate statistical distributions, their properties and some initial applications. We focus on the C++ <random> library and the Boost Statistics library. The advantage of this approach in our opinion lies in the fact that we use standardised and de-facto standard libraries. The main topics are:

• How C++ and Boost support univariate distributions.
• Black–Scholes–Merton option pricing using the C++11 error function. Computing option sensitivities.
• Generating input data to numerical processes using random number generators.
• Creating generic classes to encapsulate functionality across a wide range of distribution types in both <random> and Boost.

Much of the flexibility of the code can be attributed to features that C++ supports, for example variadic parameters, tuples and template–template parameters. We also introduce the Command design pattern (GOF, 1995) in the context of callback functions and event triggering mechanisms.

## 15.2 The Error Function and its Universality

The error function (also called the Gaussian error function) is defined by:

(15.1) In statistics this function has the following interpretation: let X be a normally distributed random variable with mean 0 and variance 1/2. Then erf(x) describes the probability of X falling in the ...

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