4 Summarizing Distributions with Expectations

Chapter Overview¶

Where We Are¶

In the last chapter we discussed strategies for visualizing functions, in particular, distribution functions. In this chapter we’ll study methods for summarizing distributions.

Where We’re Going¶

In particular, we’ll focus on expectations. Expectations are summary numbers whose values describe different aspects of a distribution. The two most famous are the expected value which summarizes the center of the distribution, and is fundamental to how we measure chance, and variance which measures how much a distribution spreads, or, how much samples from the distribution vary.

1. Section 4.1 will introduce expected values as measures of central tendency. We will:

   • Define expected values as weighted averages.

   • Contrast expected value with the median and mode of a distribution.

   • Show how to compute the expected value of $Y = g(X)$.

2. Section 4.2 will introduce a series of properties of expectation. These are extremely helpful since they make calculating expectations straightforward. We will:

   • Discuss the properties of expectation.

   • Apply the properties to compute the expectation of count variables (e.g. Binomial).

3. Section 4.3 will define variance and standard deviation in terms of expectations. We will:

   • Compare standard deviation to other measures of spread.

   • Introduce some basic properties of standard deviation.

   • Compare variance to other moments that summarize the shape of a distribution