# ST111 and ST112: Probability A & B

###### Lecturer(s)

Dr Jon Warren

* Prerequisites:* MA131 Analysis I, MA132 Foundations.

* Commitment:* 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). 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 (part A):**

1. 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.

2. Simple examples of discrete probability spaces. Methods of counting: inclusion-exclusion formula

and multinomial co-efficients. Examples including the birthday problem and coupon collecting.

3. Simple examples of continuous probability spaces. Points chosen uniformly at random in space.

4. Independence of events. Conditional probabilities. Simpsonâ€™s paradox. Bayes theorem.

5. Binomial probabilities. The law of large numbers, Poisson and Gaussian approximations and their

applications.

**Content (part B):**

6. 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.

7. Joint distributions. Independence of random variables.

8. Expectation of random variables. Properties of expectation.

9. Variance and Chebyshev's inequality. Covariance and the Cauchy-Schwartz inequality.

10. Addition of independent random variables: convolutions. Generating functions, Moment

generating functions and their use to compute convolutions.

11. Important families of distributions: Binomial, Poisson, negative Binomial, exponential, Gamma

and Gaussian. Their properties, genesis and inter-relationships.

12 The law of large numbers and the Central limit theorem.

* 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.

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

* Books:* Durrett, Elementary Probability for Applications.

Grimmett and Walsh, Probability- An Introduction.

Grimmett and Stirzaker, One Thousand Exercises in Probability

**Deadline:**

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

Resources for current ST111/112 students (restricted access)