# Chapter 2Random Variables, Probability Distributions, and Important Statistical Concepts

To understand the risk and expected return associated with an individual asset or a portfolio, one needs a way to model uncertainty. Mathematically, information about uncertainty can be summarized with probability distributions. Expected return and risk are measures describing features of these probability distributions.

This chapter reviews the concepts of random variables, discrete and continuous probability distributions, distribution summary measures, and a law in statistics called the Central Limit Theorem. An important part of the chapter is the introduction of the concept of risk measures. Some of the most widely used risk measures in portfolio management such as standard deviation, value-at-risk, and conditional value-at-risk are explained.

## 2.1 What Is a Probability Distribution?

A natural way to think of uncertainty is in terms of *scenarios*. Scenarios represent possible events that could happen. For example, the value of a stock you own but are contemplating selling may go up (one scenario) or down (another scenario) one year from now. To these scenarios, you could assign *probabilities*, which reflect your estimate of the likelihood that the scenarios will occur. For example, you estimate that the probability that the stock's value will go up is 0.30 (30%), and the probability that it will go down is 0.70 (70%).

The information contained in the scenarios and the probabilities can ...