##
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

- 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.