Chapter 4 Simulating Distributions

In this chapter we will investigate some examples that further illustrate properties of discrete and continuous random variables and their distributions, the simulation process, and Symbulate code.

Recall that the (probability) distribution of a random variable specifies the possible values of the random variable and a way of determining corresponding probabilities. The distribution of a random variable specifies the long run pattern of variation of values of the random variable over many repetitions of the underlying random phenomenon. The distribution of a random variable (\(X\)) can be approximated by

• simulating an outcome of the underlying random phenomenon (\(\omega\))
• observing the value of the random variable for that outcome (\((X(\omega)\))
• repeating this process many times
• then computing relative frequencies involving the simulated values of the random variable to approximate probabilities of events involving the random variable (e.g., \(\textrm{P}(X\le x)\)).

We will discuss distributions in more detail in the next chapter, including their mathematical representations and properties. The examples in the current chapter provide an introduction to some of the ideas. One key idea is that any distribution can be represented by a spinner. For example, the spinner on the left in Figure 2.9 corresponds to a single roll of a fair four-sided die.