Lesson 14: Continuous Random Variables

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:

  1. 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 \(X\).
  2. 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.