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Chapter 3 Probability Models

Although probability mass functions and probability density functions are studied in general it is helpful to focus on a subset of functions which describe the distribution of outcomes of broad classes of experiment. In this chapter we present some of the most widely used. Further frequently-used distributions are detailed in Chapter 11. The distributions arise in two ways, either as the probability distributions that are used in statistical modelling contexts or as the distributions that arise in statistical techniques.

Because of the strong links to statistical modelling, where interest is in the effect on the distribution of the choice of parameter value, here we use the notation $f_{X}(x;\boldsymbol{\theta})$ for $f_{X}(x)$ to emphasise the dependence of the probability on the parameters $\boldsymbol{\theta}$ of the probability model. (cf Statistics modules).

3.1 Discrete Probability Models

3.2 Uniform Distribution: ${\operatorname{\mathsf{Unif}}}(a,b)$

3.3 Exponential Distribution: $\operatorname{\mathsf{Exp}}(\beta)$

3.4 Gamma Distribution: $\operatorname{\mathsf{Gam}}(\alpha,\beta)$

3.5 The Beta Distribution: $\operatorname{\mathsf{Beta}}(\alpha_{1},\alpha_{2})$

3.6 The Weibull Distribution: $\operatorname{\mathsf{Weib}}(\alpha,\beta)$

3.7 Normal Distribution: $\operatorname{\mathsf{N}}(\mu,\sigma^{2})$

3.8 The Cauchy Distribution: $\operatorname{\mathsf{Cauchy}}$

3.9 Summary of probability distributions