Appendix A - Mathematical and statistical preliminaries

Summary

Probability laws and distributions

Sample space and events

The sample space

The sample space is the set of all possible values, or outcomes, of a realization that is not known, be it in the past, present or future. In the Bayesian probabilistic framework, every unknown quantity is treated as a random quantity.

Examples:

1. When tracking an object in 3D space, the position of the target at some point in time in the future is not known. The 3D space is the sample space.

2. The exact position of that object in the past may not be known. In many tracking situations, the exact position is never observed, only estimated. In that case, although in the past, the exact position of the target is a random quantity and the 3D space is the sample space.

3. Measurements in object tracking are the results of observations by sensing devices. They are subject to random fluctuations. Measurement errors are attached to the measurements, making them random quantities. The focus is on the errors and they are treated as random values. Their sample space is problem dependent, but often is the value space of the measurements.

The sample space is the mathematical set of all values that can be taken by an unknown quantity of interest. One of the simplest examples would be the tossing of a coin. The sample space is {H, T}, where H is the outcome of a head in the tossing and T is tail.