Probability theory and statistics

Chapter 2 : Random variables

By Lund University

Random variable is the probably the most important concept in probability theory. Based on the setup from chapter one, we give a somewhat simplified definition of a random variable. A given random variables will have associated distribution functions which will help us calculate probabilities and these will be analyzed in the second section. We then look at the most important random variable, the standard normal.

Random variables

Given an experiment with possible outcomes, a collection of events and probabilities assigned to each of these events, we are in a position to define a random variable. It is difficult to give a formally correct definition of a random variable without spending some time on measure theory which we want to avoid in this course. In this first section we will look at a simplified definition which is close to the complete one. A random variable is defined as a function, a function which will map each outcome to a real number. In order for such a function to be a random variable it must actually satisfy a condition called measurability, which we will not discuss in this course. Once we have defined a random variable, we can look at probabilities that this random variable will take certain values.

Random variable

Random variable and probabilities

Problem: Events defined by a random variable

Distribution functions

For a given random variable X, we define its cumulative distribution function (CDF) as the probability that X will take a value smaller than or equal to a given real number x. We will look at two types of random variables in this course: discrete random variables and continuous random variables. All discrete random variables have an associated probability mass function (PMF) and we will look at the relationship between the CDF and the PMF. All continuous random variables have a probability density function (PDF) and we will also look at the relationship between the CDF and the PDF for such random variables.

CDF, cumulative distribution function

Discrete random variable and probability mass function

Discrete random variable: probability mass function versus cumulative distribution function

Continuous random variable and the probability density function

The pdf and the cdf of a continuous random variable

The standard normal random variable

In this short third section, we only have one lecture dealing with the most important continuous random variable called the standard normal random variable.

Standard normal