Distributions for Counts and Proportions

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CHAPTER OUTLINE

  • 5.1 The Binomial Distributions
  • 5.2 The Poisson Distributions
  • 5.3 Toward Statistical Inference

Introduction

In Chapter 4, we learned the basic concepts of probability leading to the idea of a random variable. We found that random variables can either be discrete or continuous. In terms of discrete random variables, we explored different examples of discrete probability distributions, many of which arise from empirical observation.

For this chapter, we have set aside two important discrete distributions, binomial and Poisson, for detailed study. We will learn that these distributions relate to the study of counts and proportions that come about from a particular set of conditions. In implementing these models, there will be occasions when we need a reliable estimate of some proportion as an input. The use of an estimate leads us naturally to discuss the basic ideas of estimation, moving us one step closer to a formal introduction of inference, the topic of the next chapter.

Why are we giving special attention to the binomial and Poisson distributions? It is because the understanding of how counts and proportions behave is important in many business applications, ranging from marketing research to maintaining quality products and services.

  • Procter & Gamble states “customer understanding” as one of its five core strengths.1 Procter & Gamble invests hundreds of millions of dollars annually to conduct thousands of marketing research studies to determine customers’ preferences, typically translated into proportions. Procter & Gamble, and any other company conducting marketing research, needs a base understanding of how proportions behave.
  • When a bank knows how often customers arrive at ATMs, there is cash available at your convenience. When a bank understands the regular patterns of online logins, banks can quickly identify unusual spikes to protect your account from cybercriminals. Do you know that specialists at banks, like Bank of America and Capital One, need an understanding of the Poisson distribution in their toolkit?

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  • If you follow soccer, you undoubtedly know of Manchester United, Arsenal, and Chelsea. It is fascinating to learn that goals scored by these teams are well described by the Poisson distribution! Sports analytics is sweeping across all facets of the sports industry. Many sports teams (baseball, basketball, football, hockey, and soccer) use data to drive decisions on player acquisition and game strategy. Sports data are most often in the form of counts and proportions.