PE2-2.2 Probability Theory & Random Processes in Simulation
You can apply Bayes' theorem and work with random variables and distributions, generate random numbers and histograms, and simulate random processes to obtain power spectra of signals and noise.
The point of this unit is that a distribution you can sample is a distribution you can experiment on — simulate ten thousand trials and the law of large numbers shows you the answer before you finish the algebra. Generating a specified distribution from a uniform source is the enabling trick and worth knowing by the inverse-transform method. Histograms are estimates, not truths: bin choice changes what you see, which is a good early lesson in trusting your own plots carefully.
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Probability and Bayes' theorem
The foundations, and the theorem that lets you reason backwards from evidence. It is used directly in the detection topic later.
Random variables; clinical trials as an example
Random variables introduced through a worked example from outside engineering. The example is chosen so the mathematics is not obscured by the application.
Random numbers and random distributions
Where computer-generated randomness comes from and what distribution it has. Knowing that it is deterministic underneath matters for reproducibility.
Histograms and their bin sensitivity
Looking at a distribution by counting, and the fact that the picture changes with the bin width. It is a good early lesson in how a visualisation can mislead.
Functions of random variables
What happens to a distribution when you transform the variable. It is needed by the generation methods in the next item.
Generating random distributions
How to produce samples from a distribution you want, given samples from one you have. The inverse transform method is the one to understand properly.
Laws of large numbers
Why averages settle down, which is the theoretical backing for every simulation in this course. It also tells you how many runs you need.
Random processes and their properties
Randomness that evolves in time, and the properties that make it tractable. Stationarity and ergodicity are the two assumptions everything later relies on.
Power spectra; signals and noise
Describing a random signal by its frequency content, and the noise model the whole subject uses. It connects this topic to the communication half of the course.
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