Overview

Elegant and powerful techniques are at the fingertips of statisticians. Although difficult at first, speaking their language can be quite powerful. Probability theory provides a set of tools that can be used to quantify uncertain events. In the context of robotics, probability theory allows us make decisions in the presence of uncertainty caused by phenomena such as noisy sensor data or interaction with unpredictable humans. This section introduces a few fundamental concepts including probability, random variables, distributions, and Gaussian random vectors.

When we talk about probability, we generally talk in terms of experiments and outcomes. When an experiment is conducted, a single outcome from the set of possible outcomes for that experiment results. For example, an experiment could be flipping a coin and the set of possible outcomes is {heads, tails}.If the experiment were to take a measurement in degrees Kelvin, then the set of possible outcomes would be the interval [0, ?). An event is defined to be a subset of the possible outcomes.

Let denote the set of all possible outcomes for a given experiment, and let E be an event, i.e., . The probability of the event E occurring when the experiment is conducted is denoted Pr( E). Pr maps to the interval [0, 1]. In the example of flipping a fair coin, Pr(heads) = 0.5, Pr(tails) = 0.5, and Pr(heads ? tails) = 1. In general, the probability must obey certain properties:

1. 0 ?