Raising the bar

Chapter 1. How to Study Statistics?

Unit 1.2. Studying a definition

A definition starts with a preamble which sets a background and allows you to see the existence of objects with distinct properties. Most of the time the preamble is not even mentioned but there are cases when it contains a complex logical argument. After you understand the preamble understanding the definition proper -- that is giving the name -- becomes easy. Studying a definition is concluded by considering possible equivalent definitions and deriving immediate consequences.

Example

(1) Preamble. Let us consider natural numbers (these are numbers $latex 1,\,2,\,3,\,...$ that naturally arise when counting things). [Kronecker is reported to have said, "God created the natural numbers; all the rest is the work of man."] We can notice that some of them are divisible by 2 (for example, $latex 4/2 = 2$ is an integral number) and others are not (e.g., $latex 3/2$ is a fractional number).

(2) Definition proper. The natural numbers that are divisible by 2 ($latex 2,\,4,\,6,\,...$) are called even. The natural numbers that are not divisible by 2 ($latex 1,\,3,\,5,\,...$) are called odd.

(3) Immediate consequences. (i) Any natural number is either even or odd. (ii) If a natural number is even and divisible by 3 at the same time, then it is divisible by 6.

When a definition is not possible

Some basic objects cannot be defined or their precise definitions are too complex for this course. In such cases it is better to give equivalent (descriptive) names or simply list the objects we name (as it has been done in case of natural numbers).

Example. The following names are used interchangeably: set = collection = family = array.

Example. Natural numbers are members of the set $latex \mathbb{N} = {1,\,2,\,3,\,...}$. Integer numbers are members of the set $latex \mathbb{Z} = {0,\,\pm 1,\,\pm 2,\,...}$. In some applications (see Units 5.15 and 6.9) we shall need nonnegative integer numbers listed in $latex \mathbb{Z}_+ = {0,\,1,\,2,\,...}$.

Example. The next type of numbers we need, real numbers, cannot be put in a list. As a convenient working definition, we call real numbers those numbers that can be written in a decimal form. For example, the famous number $latex \pi$ is, by definition, the ratio of the length of a circumference to its diameter. Its decimal form is, up to approximation, $latex \pi = 3.1415926...$ Integer numbers fall into this category: for example, $latex 1 = 1.000...$ The set of all real numbers is denoted $latex \mathbb{R}$.

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