Distributions, Moments, and Errors

Abstract

(a) Sample versus population distributions; (b) Multi-variate distributions; (c) Summarising quantities for distributions; (d) Expectations and moments; (e) Transformation of probability distributions; (f) Error analysis.

In the first chapter we began to look at how to reason in the presence of uncertainty. This led us to the idea of a probability distribution P(X) or p(x) for a random variable. This might for example represent the distribution of molecular velocities in a hot gas. In order to perform reasoning based on such distributions, we need to know how to characterise and manipulate them. What is the typical value? What is the spread? How do we deal with multiple variables? If we know the distribution for a variable, can we deduce the distribution for a related variable? In this chapter we look at some mathematical techniques to answer these questions. This will set us up for dealing with real world distributions, and understanding how to perform statistical inference. First however, we need to carefully distinguish between theoretically expected and empirically observed distributions.