S4-5.1 Probability & Random Variables

Work through these

• Joint, conditional and total probability; Bayes' theorem; independence

  Conditional probability, the total probability rule and Bayes' theorem are the machinery for updating a belief when evidence arrives, and independence is the assumption that makes many problems tractable. This is the load-bearing item of the whole subject.

  NPTEL: Probability and Stochastic Processes · Course

• Concept of a random variable

  A random variable attaches a number to the outcome of a random experiment, which is what lets probability be done with calculus rather than counting. The idea is small and everything after it depends on the definition being precise.

• Continuous distributions: uniform, exponential, Gaussian, Rayleigh

  Four continuous distributions chosen for engineering reasons: Gaussian because noise behaves that way, Rayleigh because fading signal envelopes do, exponential for waiting times, uniform as the base case. Recognising which one a situation calls for is the skill.

• Discrete distributions: binomial, Poisson

  Binomial counts successes in a fixed number of trials; Poisson counts arrivals in an interval when they are rare and independent. Both turn up constantly in traffic, errors and packet arrivals.

• Conditional and joint distribution and density functions

  When two random quantities are involved, the joint density describes them together and the conditional density describes one given the other. This is the groundwork for correlation and for every later result about processes.