Chapter Overview¶
Where We Are¶
Chapter 1 established the essential rules of chance. These all stem from the three axioms: nonnegativity, normalization, and additivity (see Section 1.3). The rules are essential since they:
Establish the constraints all probability models obey
Provide algebra rules that make it easier to compute chances
In general, our computational strategy comes in three parts:
This scheme is missing an initial step. It does not tell us how to pick our chance model. It only tells us which proposed models are valid. Leaving Chapter 1, our only concrete chance model is probability as proportion, which only applies if outcomes are equally likely (Section 1.2).
In a sense, we’ve set the stage, but have not introduced the players. This chapter introduces our main characters.
Where We’re Going¶
The chapter will introduce the two basic categories of chance models we will study this semester. It will:
Define random variables and introduce the idea of a distribution function (Section 2.1)
Relate random variables to measurements associated with random outcomes.
Show how a distribution function can represent a probability measure.
Distinguish types of distribution functions (cumulative or local).
Introduce discrete random variables and their distribution function, the probability mass function (PMF) (Section 2.2)
Show how to derive a PMF from the description of a random process
Introduce three key models: the (1) Bernoulli, (2) Geometric, and (3) Binomial.
Study the shape of the associated PMF’s.
Examine how parameters specify models.
Introduce continuous random variables and their distribution function, the probability density function (PDF) (Sections 2.3 and 2.4)
Study the probability of exact events to distinguish possible and probable.
Interrogate what we mean when we assume continuity
Distinguish a density function (PDF) from a (PMF)
Show how to compute chances given a PDF
Introduce four key models: the (1) Uniform, (2) Exponential, (3) Pareto, and (4) Normal.
Study the shape of the associated PDF’s and contrast modeling by shape to modeling by story.
Examine how parameters specify models.
A chapter summary is available in Section 2.5