3 Visualizing Functions and Models

Chapter Overview¶

Where We Are¶

Chapters 1 and 2 introduced the two main questions we’ll focus on in this course. These are:

1. Computing chances. For example, find the probability of event $E$. Or, what’s the chance that I see 7 successes in 10 trials?

2. Picking a chance model. For example, are all outcomes equally likely? Is $X$ geometric or binomial? Is it continuous or discrete? What distribution function should I use?

We can’t answer the first question without answering the second. Chapter 1 provided rules for computing chances given a chance model. Chapter 2 showed three ways to pose a chance model (PMF, CDF, PDF), provided examples when we could derive a chance model from a process (see the Geometric and Binomial examples in Section 2.2), and introduced examples where it made more sense to simply pick a distribution function based on its shape (Section 2.4).

Where We’re Going¶

To either pose a chance model by selecting a function with a desired shape, or to understand a chance model by converting from a function to a shape, we will need to get good at visualizing functions. That’s the “graphical” in the course title.

This chapter is all about ways to summarize functions, describe their properties, and use those descriptions to go back and forth between visuals and formula. Building this instinct in your mind’s eye is foundational if you want to get good at picking models, or at understanding their behavior.

1. Section 3.1 will introduce ways to categorize functions based on their global properties. These are:

   • symmetry

   • monotonicity

   • nonnegativity

   • convexity/concavity

   • asymptotics

   • This chapter will also discuss ways to visualize transformations and combinations of functions. We will discuss:

   • scaling and translation

   • inversion

   • function addition and products

   • function composition

   • We will provide interactive tool kits to help you practice categorizing functions, and recognizing how they change when transformed or combined.

   • We will focus on examples relevant to probability.

2. Section 3.2 will introduce ways to characterize functions by studying their local properties at strategically selected points. These are:

   • Local linear approximation. In essence, taking derivatives and using them to approximate the function.

   • Finding critical points (maxima, minima, inflection points)

   • Using second derivatives to check concavity/convexity

As you can see by this list, this chapter is less conceptual than the first two. We’ve done the conceptual work. It’s time to start developing some practical skills. If you’ve seen thse skills before, use this chapter as a refresher, and, if you find it all familiar, go back and spend your time on ideas you found confusing in the first two chapters.