# ST112 Probability B

## Resources

###### Lecturer(s): Dr Jon Warren

* Prerequisites:* MA131 Analysis I, MA132 Foundations.

* Leads to:* ST104 Statistical Laboratory, ST220 Introduction to Mathematical Statistics, ST202 Stochastic Processes, MA3H2 Markov Processes and Percolation Theory, and to numerous statistical, probabilistic, operational research and econometric courses.

* Commitment: *This module runs in Term 2.

- ST111 - 15 hours of lectures, 2 tutorial hours (week 3 and week 5)
- ST112 - 15 hours of lectures, 2 tutorial hours (week 7 and week 9).

* 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 (part A):**

- Experiments with random outcomes: the notions of events and their probability. Operations with sets and their interpretation. The addition law and axiomatic definition of a probability space.
- Simple examples of discrete probability spaces. Methods of counting: inclusion-exclusion formula and multinomial co-efficients. Examples including the birthday problem and coupon collecting.
- Simple examples of continuous probability spaces. Points chosen uniformly at random in space.
- Independence of events. Conditional probabilities. Simpsonâ€™s paradox. Bayes theorem.
- Binomial probabilities. The law of large numbers, Poisson and Gaussian approximations and their applications.

**Content (part B):**

- The notion of a random variable and its distribution. Examples in both discrete and continuous settings. Probability mass functions and density functions. Cumulative distribution functions.
- Joint distributions. Independence of random variables.
- Expectation of random variables. Properties of expectation.
- Variance and Chebyshev's inequality. Covariance and the Cauchy-Schwartz inequality.
- Addition of independent random variables: convolutions. Generating functions, Moment generating functions and their use to compute convolutions.
- Important families of distributions: Binomial, Poisson, negative Binomial, exponential, Gamma and Gaussian. Their properties, genesis and inter-relationships.
- The law of large numbers and the Central limit theorem.

**Books:**

- Durrett, Elementary Probability for Applications.
- Grimmett and Walsh, Probability- An Introduction.
- Grimmett and Stirzaker, One Thousand Exercises in Probability
- Sheldon Ross, A first course in Probability.

* Assessment: *10% assessed work (during term 2) and 90% written examination (in term 3).

**Deadlines:**

- ST111 assignments are due on Tuesdays of weeks 4 and 6
- ST112 assignments are due on Tuesdays of weeks 8 and 10

* Feedback: *Feedback on your assignments will be given within 2 weeks of submission.