Chapter Overview¶
This chapter focuses on methods for approximating functions. We will start by approximating functions with polynomials, then via Taylor series. After developing methods for approximating exponentials, logarithms, and factorials, we will combine our tools to study distributions that are produced by limiting arguments. These include two of the most important distributions in probability, the Poisson distribution and the normal distribution.
Section 6.1 will introduce Taylor series as a tool for approximating functions.
We will derive the Taylor series for the exponential and logarithm
We will practice checking convergence using the ratio test
We will discuss the accuracy of polynomial approximations produced by truncating the Taylor series
Section 6.2 will use Taylor series to approximate exponentials and logarithms.
We will prove the limiting definition of the exponential, then show how to use it to approximate probabilities with exponential functions
We will use exponential approximations to derive the exponential distribution from a generating process
Section 6.3 will introduce Stirling’s approximation for factorials.
Section 6.4 will combine our tools to study limiting distributions.
We will develop the Poisson distribution as a limit of binomials and relate it to the process that produces exponential random variables
We will illustrate the central limit theorem by deriving the normal distribution as a limit of binomials.
This is the most formula-heavy chapter in the book. The results are beautiful, but they’ll take some hard work.
Section 6.1 provides the most useful general tool in the chapter, while Sections 6.2 and 6.4 provide the greatest probability insight. They will provide our first example derivations of continuous distributions from processes that do not start with a uniform assumption and will introduce the two most important distributions in applied probability.