11Commonly Used Functions

There is a wide range of mathematical functions used in statistics. The reader may not be familiar with many of them. The symbols used and the language involved can be daunting. Their value is that they allow us to simply document a complex combination of basic mathematical operations. They are analogous to using n! to replace 1 × 2 × 3 × … × n. This chapter includes those most commonly used. With a little practice, the reader will quickly become comfortable using them. Indeed, they are becoming commonplace in spreadsheet packages.

11.1 Euler’s Number

The reader is likely to be familiar with a number of mathematical constants used frequently in statistical formulae. For example, we know π is the ratio of the circumference of a circle to its diameter. Another is Euler’s number (e) but less well‐known is its definition, which is

(11.1)images

It is important because similar functions can appear in the derivation of a PDF or CDF and so lead to the inclusion of an exponential function. For example,

This convergence is illustrated in Figure 11.1.

n vs. (1 + λ/n)n illustrating the convergence to powers of Euler’s number displaying three ascending curves for e2 = 7.389, e1 = 2.718, and e–2 = 0.135.

Figure 11.1 Convergence to powers of Euler’s number

Equation (11.2) is often the basis for the limit of other ...

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