# 5: Random Variables


• 5.1: Introduction
The graph of a continuous probability distribution is a curve. Probability is represented by area under the curve. The curve is called the probability density function (abbreviated as pdf).
• 5.2: Probability Distribution Function (PDF) for a Discrete Random Variable
A discrete probability distribution function has two characteristics: Each probability is between zero and one, inclusive. The sum of the probabilities is one.
• 5.3: Mean or Expected Value and Standard Deviation
The expected value is often referred to as the "long-term" average or mean. This means that over the long term of doing an experiment over and over, you would expect this average. This “long-term average” is known as the mean or expected value of the experiment and is denoted by the Greek letter μμ . In other words, after conducting many trials of an experiment, you would expect this average value.
• 5.4: Continuous Probability Functions
The probability density function (pdf) is used to describe probabilities for continuous random variables. The area under the density curve between two points corresponds to the probability that the variable falls between those two values. In other words, the area under the density curve between points a and b is equal to P(a<x<b)P(a<x<b) . The cumulative distribution function (cdf) gives the probability as an area.
• 5.5: Exercises
These are homework exercises to accompany the Textmap created for "Introductory Statistics" by OpenStax.
• 5.E: Continuous Random Variables (Exercises)
These are homework exercises to accompany the Textmap created for "Introductory Statistics" by OpenStax.
• 5.E: Discrete Random Variables (Exercises)
These are homework exercises to accompany the Textmap created for "Introductory Statistics" by OpenStax.