**Aims & Learning Objectives:**
**Aims: **To introduce some fundamental topics in probability theory including conditional expectation and the three classical limit theorems of probability.
To present the main properties of random walks on the integers, and Poisson processes.
**Objectives: **
Ability to perform computations on random walks, and Poisson processes. Ability to use generating function techniques for effective calculations.
Ability to work effectively with conditional expectation.
Ability to apply the classical limit theorems of probability.
**Content: ** Revision of properties of expectation and conditional probability. Conditional expectation.
Chebyshev's inequality. The Weak Law.
Statement of the Strong Law of Large Numbers.
Random variables on the positive integers.
Probability generating functions.
Random walks expected first passage times.
Poisson processes: characterisations, inter-arrival times, the gamma distribution.
Moment generating functions. Outline of the Central Limit Theorem.
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