4.11 One ring spinner to rule them all?

In the examples in this section we used different spinners to represent different distributions. However, all of the examples assumed the same generic spinner: the needle was infinitely precise and “equally likely” to land on any value on the axis around the spinner. We modeled different distributions simply by changing the values on the axis.

Consider the standard continuous spinner in Figure 2.2, corresponding to a Uniform(0, 1) distribution. By relabeling the axes on this spinner, we could have constructed the spinners for any of the other examples.

For example, to obtain the spinner in the middle of Figure 2.9, corresponding to a weighted four-sided die, start with the Uniform(0, 1) spinner and map

• The range (0, 0.1] to 1,
• The range (0.1, 0.3] to 2,
• The range (0.3, 0.6] to 3,
• The range (0.6, 1] to 4

Then the probability that the Uniform(0, 1) spinner lands in the range (0.3, 0.6] is 0.3, so the spinner resulting from this mapping would return a value of 3 with probability 0.3. (The probability of the infinitely precise needle landing on a specific value like 0.3 (that is, \(0.300000000\ldots\)) is 0, so it doesn’t really matter what we do with the endpoints of the intervals.)

For non-uniform values on a continuous scale, we could construct a spinner according to the distribution of interest by rescaling and stretching/shrinking the axis of the Uniform(0, 1) spinner to correspond to intervals of larger/smaller probability. For example, if we want to simulate values according to the distribution illustrated in Figure 4.6 we could start with the Uniform(0, 1) spinner and then transform the axis values \(u \mapsto -\log(1-u)\) to obtain the spinner in Figure 4.7. As discussed in Section 4.8, the spinner in Figure 4.7 generates values which follow the distribution is Figure 4.6.

In Section 4.8 we started with the transformation \(u\mapsto -\log(1-u)\) of the Uniform(0, 1) spinner and saw what distribution the transformed values followed via simulation. But what about the reverse question: given a particular distribution, how do we find the transformation of Uniform(0, 1) that will generate values according to the specified distribution? We will return to this question later.

The only example in this section where a Uniform(0, 1) spinner could not be used was the SAT example in Section 4.10, where we described a “globe” for simulating values. However, we will see later that we actually can use a Uniform(0, 1) to generate a pair of SAT scores, but we will need to suitably transform the results of two spins.

Through the examples in this chapter we have seen that, in principle, we can start with a Uniform(0, 1) spinner and via a suitable transformation of the axis — and possibly multiple spins — generate values according to any distribution of interest. This is the idea behind what is sometimes referred to as “universality of the uniform”, or what we like to call “one spinner to rule them all”, and we will explore it further later.