# ST115: Introduction to Probability

###### Lecturer(s): Dr Zorana Lazic

Important: This module is for students from the Statistics department only. Students from other departments should take ST111 Probability A and ST112 Probability B.

Prerequisite(s):
MA137 Mathematical Analysis, MA138 Sets and Numbers

Commitment: 3 lectures/week, 1 tutorial/fortnight, 1 exercise class/week. This module runs in Term 2.

Aims: To lay the foundation for all subsequent modules in probability and statistics, by introducing the key notions of mathematical probability and developing the techniques for calculating with probabilities and expectations.

Content:

 1 Experiments with random outcomes: the notions of events and their probability. Operations with sets and their interpretation. The addition law. 2 Simple examples of discrete probability spaces. Methods of counting: inclusion-exclusion formula and Binomial co-efficients. 3 Simple examples of continuous probability spaces. Points chosen uniformly at random in space. 4 Independence of events. Conditional probabilities. Bayes theorem. 5 The notion of a random variable. Examples in both discrete and continuous settings. Indicator random variables. 6 The notion of the distribution of a random variable. Probability mass functions and density functions. Cumulative distribution functions. 7 Expectation of random variables. Properties of expectation. 8 Mean and variance of distributions. Chebyshev's inequality. 9 Independence of random variables. Joint and conditional distributions. Covariance. Cauchy-Schwartz inequality. 10 Addition of independent random variables: convolutions. Generating function and use to compute convolutions. 11 Important families of distributions: Binomial, Poisson, negative Binomial, exponential, Gamma and Gaussian. Their properties, genesis and inter-relationships.

Then the following topics to be covered in second year may be introduced at the end of this module.

Sequences of random variables. Convergence in probability and distribution. The Weak law of large numbers. generating functions and relationship with convergence. The Central limit theorem.

Books:

• Ross, A first course in probability, Prentice Hall, 1994
• Pitman, Probability, Springer texts in Statistics
• Suhov and Kelbert, Probability and Statistics by Example: Basic probability and Statistics.

Assessment: 90% by 2 hr examination, 10% coursework.

Deadlines: Assignment 1: week 3, Assignment 2: week 5, Assignment 3: week 7, Assignment 4: week 9.

Feedback: You will hand in answers to selected questions on the fortnightly exercise sheets. Your work will be marked and returned to you in the tutorial taking place the following week when you will have the opportunity to discuss it.