Monday, January 21, 2008

Discrete Random Variables (Section 2.1)

My philosophy is that class time should be used to facilitate learning and not simply to reiterate concepts that the book already explains adequately. I readily acknowledge that I am still learning how to accomplish such a feat, especially to help students overcome the tendency to avoid reading their textbook.

As we leave chapter 1 where we learned basic ideas of the probability of events, we begin chapter 2 where we will focus on a family of random variables of the discrete type.
Comparing Definition 2.1-1 with the definition I game in my first week slides, you should notice that there is a distinction between the outcome space of the experiment and the space of the random variable. The outcome space should represent the detailed description of the experiment, while the space of the random variable is the range of the random variable (as a function of the outcome or sample space). Pages 58 and 59 provide an important philosophical guide for what we are trying to accomplish and point out that observations might help us to estimate the probabilities associated with the random variable. However, we can often use basic assumptions to create a mathematical model for these probabilities. This chapter introduces a number of models that describe discrete random behavior.

Definition 2.1-2 is very important, introducing the definition of the probability mass function. Problem 3 in the textbook helps test if you understand the basic ideas. One of the major points you need to remember is that for discrete type random variables, properties (b) and (c) compute probabilities using summations. When we get to continuous type random variables, the corresponding properties will replace summation with integration.

In addition to basic principles (probability mass function (mathematical model' prediction) vs relative frequency (statistical estimate), bar graph vs histogram), we meet the first model for a random variable---the hypergeometric distribution---which describes choosing n objects from a total collection of two types of objects. The model is created by considering exactly the types of calculations used in chapter 1, by counting how many ways to select n objects from a total of N objects (denominator) and then also counting how many ways to choose x of the first type and n-x of the second type (numerator).