Probability and Random Variables

Understanding uncertainty relies heavily on the concepts of random variables and probability. Random variables are used to represent quantities that can take on different values based on a particular probability distribution. They play a crucial role in modeling uncertainty and randomness in diverse areas such as finance, physics, and engineering. Probability theory provides a framework for modeling the likelihood of various events occurring based on the underlying distribution of the random variables involved.

Outline

In this appendix, we will discuss the following topics:

• A random variable is a variable \(X\) that takes on different values \(x\) according to the outcome of a random event or process. There are two types of random variables: discrete random variables and continuous random variables. Discrete random variables can take on a countable number of distinct values, while continuous random variables can take on any value in a continuous range.

• Probability measures the likelihood that a particular event or outcome will occur and is commonly used to quantify uncertainty in various fields, such as science, engineering, economics, and finance. For a discrete random variable, the likelihood that \(X=x\) is described by a Probability Mass Function (PMF) and a Probability Density Function (PDF) for continuous random variables.