Lesson 15: Exponential, Gamma and Chi-Square Distributions

Overview Section

In this Chapter, we investigate the probability distributions of continuous random variables that are so important to the field of statistics that they are given special names. They are:

  • the uniform distribution (Lesson 14)
  • the exponential distribution
  • the gamma distribution
  • the chi-square distribution
  • the normal distribution

In this lesson, we will investigate the probability distribution of the waiting time, \(X\), until the first event of an approximate Poisson process occurs. We will learn that the probability distribution of \(X\) is the exponential distribution with mean \(\theta=\dfrac{1}{\lambda}\). In this lesson, we investigate the waiting time, \(W\), until the \(\alpha^{th}\) (that is, "alpha"-th) event occurs. As we'll soon learn, that distribution is known as the gamma distribution. After investigating the gamma distribution, we'll take a look at a special case of the gamma distribution, a distribution known as the chi-square distribution.

Objectives

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

  • To learn a formal definition of the probability density function of a (continuous) exponential random variable.
  • To learn key properties of an exponential random variable, such as the mean, variance, and moment generating function.
  • To understand the steps involved in each of the proofs in the lesson.
  • To be able to apply the methods learned in the lesson to new problems. To understand the motivation and derivation of the probability density function of a (continuous) gamma random variable.
  • To understand the effect that the parameters \(\alpha\) and \(\theta\) have on the shape of the gamma probability density function.
  • To learn a formal definition of the gamma function.
  • To learn a formal definition of the probability density function of a gamma random variable.
  • To learn key properties of a gamma random variable, such as the mean, variance, and moment generating function.
  • To learn a formal definition of the probability density function of a chi-square random variable.
  • To understand the relationship between a gamma random variable and a chi-square random variable.
  • To learn key properties of a chi-square random variable, such as the mean, variance, and moment generating function.
  • To learn how to read a chi-square value or a chi-square probability off of a typical chi-square cumulative probability table.
  • To understand the steps involved in each of the proofs in the lesson.
  • To be able to apply the methods learned in the lesson to new problems.