##
Overview
Section* *

A **continuous random variable** differs from a discrete random variable in that it takes on an uncountably infinite number of possible outcomes. For example, if we let \(X\) denote the height (in meters) of a randomly selected maple tree, then \(X\) is a continuous random variable. In this lesson, we'll extend much of what we learned about discrete random variables to the case in which a random variable is continuous. Our specific goals include:

- Finding the probability that \(X\) falls in some interval, that is finding \(P(a<X<b)\), where \(a\) and \(b\) are some constants. We'll do this by using \(f(x)\), the probability density function ("p.d.f.") of \(X\), and \(F(x)\), the cumulative distribution function ("c.d.f.") of \(X\).
- Finding the mean \(\mu\), variance \(\sigma^2\), and standard deviation of \(X\). We'll do this through the definitions \(E(X)\) and \(\text{Var}(X)\) extended for a continuous random variable, as well as through the moment generating function \(M(t)\) extended for a continuous random variable.

## Objectives

Upon completion of this lesson, you should be able to:

- To introduce the concept of a probability density function of a continuous random variable.
- To learn the formal definition of a probability density function of a continuous random variable.
- To learn that if \(X\) is continuous, the probability that \(X\) takes on any specific value \(x\) is 0.
- To learn how to find the probability that a continuous random variable \(X\) falls in some interval \((a, b)\).
- To learn the formal definition of a cumulative distribution function of a continuous random variable.
- To learn how to find the cumulative distribution function of a continuous random variable \(X\) from the probability density function of \(X\).
- To learn the formal definition of a \((100p)^{th}\) percentile.
- To learn the formal definition of the median, first quartile, and third quartile.
- To learn how to use the probability density function to find the \((100p)^{th}\) percentile of a continuous random variable \(X\).
- To extend the definitions of the mean, variance, standard deviation, and moment-generating function for a continuous random variable \(X\).
- To be able to apply the methods learned in the lesson to new problems.
- To learn a formal definition of the probability density function of a continuous uniform random variable.
- To learn a formal definition of the cumulative distribution function of a continuous uniform random variable.
- To learn key properties of a continuous uniform random variable, such as the mean, variance, and moment generating function.
- To understand and be able to create a quantile-quantile (q-q) plot.
- To understand how randomly-generated uniform (0,1) numbers can be used to randomly assign experimental units to treatment.
- To understand how randomly-generated uniform (0,1) numbers can be used to randomly select participants for a survey.