This document is my compiled notes for MTH 410: Statistics at TMU. All information comes from my professor's lectures and the textbook Applied Statistics and Probability for Engineers by Douglas C. Montgomery, et al..

Adam Szava - 2tor.ca

S2022

Chapter 2: Probability

2.1: Sample Spaces and Events

All of the fields of probability exist because, in real life, there are so many factors that take part in the outcome of an experiment that we can never consider and model them all. Our goal is to understand, quantify, and model the type of variations we often encounter in such a way that our results are not invalidated by the variation.

<aside> ➕ Definition 2.1.1 (Random Experiment)

An experiment that can result in different outcomes, even though it is repeated in the same manner every time, is called a random experiment.

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An experiment may input something into a system, and measure its output as a function of that input, controllable variables, and un-controllable and changing variables (called noise).

We want to study the outcomes of experiments, but this already poses some complications. First, let’s formally define a sample space:

<aside> ➕ Definition 2.1.2 (Sample Space)

The set of all possible outcomes of a random experiment is called the sample space of the experiment denoted $S$.

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We may want to consider an experiment where we measure the growth of a plant in different soil (after a set amount of time). Possible sample spaces include:

• If we consider the length of the plant as the outcome: $S=\mathbb{R}^+$
• If we know the plants will grow to a least a height of $1$: $S=\{ x \in \mathbb{R} \mid x\geq1 \}$
• If we consider what category the plants fall into as the outcome: $S=\{ \text{short, medium, tall} \}$
• If we consider whether or not the plants meet some height threshold to be the outcome: $S = \{ \text{yes, no} \}$

As you can see the sample space is decided upon by the experimenter, and can either be continuous or discrete.

<aside> ➕ Definition 2.1.3 (Discrete and Continuous Sample Spaces)

A sample space is discrete if it consists of a finite or countable infinite set of outcomes. A sample space is continuous if it contains an interval of real numbers.

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Sometimes we also describe sample spaces as tree diagrams. Consider the example of three messages sent in a telecommunication system which can each either be late or on time. The possible outcomes of sending the three messages in sequence can be represented by tracing a line down from the root of the tree to the leaves.