Many statistical tests exploit the properties of the probability distributions of random variables. This chapter provides an introduction to some of the most important probability distributions, and lays the groundwork for the statistical tests introduced in Chapter 4.

Distributions

When we count how often a word is used, or when we measure the duration of a vowel, we carry out a statistical experiment. The outcome of such a statistical experiment varies each time it is carried out. For instance, the frequency of a word (the outcome of a counting experiment) will vary from text to text and from corpus to corpus, and similarly the length of a given vowel (the outcome of a measuring experiment) will vary from syllable to syllable and from word to word. For a given random variable, some outcomes may be more likely than others. The probability distribution of a random variable specifies the likelihood of the different outcomes. Random variables fall into two important categories. Random variables such as frequency counts are discrete (with values that are integers), random variables such as durational measurements are continuous (with values that are reals). We begin by introducing two discrete distributions.

Discrete distributions

The celex lexical database (Baayen et al., 1995) lists the frequencies of a large number of English words in a corpus of 18.6 million words. Table 3.1 provides these frequencies for four words, the high-frequency definite article the, the medium-frequency word president, and two low-frequency words, hare and harpsichord.