5 Comparing Tails

Chapter Overview¶

Where We Are¶

In the last two chapters we discussed methods for summarizing distributions by identifying key attributes of the distribution. So far our basic list of characteristics include global properties, like symmetries, interesting points, like modes (peaks of the distribution) or extreme values, and summary averages (expected values and standard deviation).

Most of these characteristics describe typical outcomes, or are used to find intervals that contain most of the probability mass. For instance, it is common practice to summarize a distribution with an interval, centered at the expected value, with width equal to 4 or 6 standard deviations. An interval of the form $\mathbb{E}[X] \pm k \text{SD}[X]$ will always contain at least $1 - (1/k)^2$ of the total probability mass. So, the interval $\mathbb{E}[X] \pm 2 \text{SD}[X]$ always contains at least 75% of the total probability mass. Then, instead of providing all the detail of the PMF or CDF, we can simply report the bounds of the interval. Like any summary, this approach discards information to make communication simpler.

In particular, it discards information about the tails of the distribution in order to focus on typical outcomes.

Where We’re Going¶

This chapter focuses on rare events. To completely summarize a distribution, we should describe the typical outcomes and the rate with which probability decays to zero for extremely unusual outcomes.

1. Section 5.1 will introduce tails by example.

   • The tail of a distribution is the part of the distribution associated with rare, or extreme events. Picture a bell curve. The tail is the bit of the bell curve far from its peak, where the distribution approaches zero.

   • We usually describe tails by focusing on the rate at which the distribution converges to zero for extreme events (e.g. very large or very small $x$).

   • We will study the tail behavior of the geometric distribution and contrast to distributions whose tails obey a power law.

2. Section 5.2 will introduce asymptotic rates.

   • We will compare rates using l’Hopital’s rule.

   • We will learn to use $\mathcal{O}$ notation to describe, and log-log plots to visualize, asymptotic rates.

3. Section 5.3 will introduce convergence tests for series.

   • We will study the integral, direct, and limit comparison tests.

4. Section 5.4 will categorize a list of important distributions according to the rates at which their tails decay.