6: Probability and Distributions
- Page ID
- 45055
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Probability is how likely the occurrence of some event is. Thus, an important concept to appreciate is that in many cases, like R.A. Fisher’s Lady tasting tea analogy, we can count in advance all possible outcomes of an experiment. On the other hand, for many more experiments, we cannot count all possible outcomes of the sample space, either because they are too numerous or simply unknowable. In such cases, applying theoretical probability distributions allow us to circumvent the countability problem. Whereas empirical probability distributions are frequency counts of observations, theoretical probabilities are based on mathematical formulas.
Much of classical inferential statistics, especially the kind one finds in introductory courses like ours, are built on probability distributions. ANOVA, t-tests, linear regression, etc., are parametric tests and assume errors are distributed according to a particular type of distribution, the normal or Gaussian distribution.
A probability distribution is a list of probabilities for each possible outcome of a discrete random variable in an entire population. Depending on the data type, there are many classes of probability distributions. In contrast, probability density functions are used to for continuous random variables. This chapter begins with basics of probability, then gently introduces probability distributions. In the other sections of this chapter we describe several probability density functions. Emphasis is placed on the normal distribution, which underlies most parametric statistics.
- 6.1: Some preliminaries
- Some preliminary principles, including definitions of probability and risk, probability of multiple events, empirical vs. theoretical probability, and skepticism of statistical findings.
- 6.2: Ratios and probabilities
- Comparisons of rates, ratios, proportions, and indexes. In-depth example of how statistics can be applied to a real-world comparison: are cars or airplanes a safer mode of travel?
- 6.3: Combinations and permutations
- Definitions and calculations for combinations and permutations of events.
- 6.4: Types of probability
- Further discussion of the types of probability, including discrete vs. continuous and theoretical vs. empirical.
- 6.5: Discrete probability distributions
- Applications, calculations, and R calculations of binomial and hypergeometric discrete probability distributions.
- 6.6: Continuous distributions
- Discussion of the Law of Large Numbers, the Central Limit Theorem, and the normal distribution and its importance in classical statistics.
- 6.7: Normal distribution and the normal deviate
- Use of the Z score, or normal deviate, to normalize any given normal distribution to the standard normal distribution. Applications of this technique in data sets, including a worked example.
- 6.8: Moments
- Moments as a way to describe the shape of a distribution, and their relationship to the expected value of the probability function. Calculating the 4 moments by hand and by using R.
- 6.9: Chi-square distribution
- The chi-square distribution as a means for testing the statistical significance of categorical variables.
- 6.10: t-distribution
- Student's t-distribution and its calculation and use cases.
- 6.11: F-distribution
- The F distribution and its use as the null distribution of the ANOVA test statistic.