2017/18, Term 2, The University of Manchester

Lecturer: Korbinian Strimmer

**Aims and Syllabus:**

The course is the second half of MATH19842: Mathematics OF2 and aims to provide a basic course in probability theory for foundation year students.

- Probability, experiments, outcomes.
- Sample spaces, events, complements and unions/intersections of events.
- Evaluation of probabilities on finite sample spaces.
- Conditional probability and independence.
- Discrete and continuous random variables.
- Probability distribution functions.
- Probability as the limit of relative frequencies.
- Mean and sample mean. Variance and sample variance.
- Standard distributions: Binomial, poisson and normal distributions with applications.

**Resources:**

There are two sets of lecture notes by Walton and Tso, respectively, on which this course is based. In addition, I recommend the first few chapters of the text book by Dekking et al. - see below for the respective links. In addition, you are encouraged to study further material on the foundations of probability, e.g. the "Random" virtual laboratories in proability and statistics:

- N. Walton. 2017. Probability OF2 Lecture Notes - PDF available on Blackboard.
- M. Tso. 2015. Probability OF2 Lecture Notes - PDF available on Blackboard.
- F. M. Dekking et al. 2005. A Modern Introduction to Probability and Statistics: Understanding Why and How - PDF available on Springer Link
- K. Siegrist. 2017. Random (formerly: Virtual Laboratories in Probability and Statistics). http://www.randomservices.org/random/.

**Computing:**

Application of probabilistic methods requires computing and statistical programming. The "lingua franca" of statistical computing is the computer language R. During the lecture I will occasionally make use of R, e.g., for random number generation and for visualisation. If you are curious please visit the R homepage and download the program and play with it! However, R is not just only useful for learning statistics, it is indeed a very powerful software that is widely used across the world in many settings, both academic and commercial. Note that R is free software. A platform-independent graphical user interface is provided by R Studio.

**History of Probability and Statistics:**

Probability has a history that goes back to (at least) the 15th century, but the modern axiomatic foundation of probability was only established in the 20th century. If you would like to better understand the context and origins of probability and statistics you may wish to read Fienberg's 1992 essay on A Brief History of Statistics in 3 1/2 Chapters, and browse biographical information on figures from the history of probability and statistics.

**Timetable and Contents:**

**Lectures:** Tuesday 5pm-6pm; Wednesday 11am-12pm.

**Tutorials:** Tuesday 4pm-5pm.

See Blackboard for further information.

Term Week | Date | Type | Content | Book chapter |
---|---|---|---|---|

6 | 6 March 2018 | Lecture 1 | Probability, uncertainty, randomness, sample space, elementary events, composite events, Kolmogorov axioms. | Chap. 2 |

6 | 7 March 2018 | Lecture 2 | Set theory and operations on sets (union, intersection, complement) and application to computing probabilities. | Chap. 2 |

7 | 13 March 2018 | Tutorial 1 | Example sheet 1 and Solutions 1. | |

7 | 13 March 2018 | Lecture 3 | Permutations, factorial, binomial coefficient. Conditional probability, multiplication formula, independence. | Chap. 3 |

7 | 14 March 2018 | Lecture 4 | More on conditional probability, law of total probability, inversion of conditioning, application to medical testing. | Chap. 3 |

8 | 20 March 2018 | Tutorial 2 | Example sheet 2 and Solutions 2. | |

8 | 20 March 2018 | Lecture 5 | Concept of random variables and their distribution, notation, discrete vs. continuous random variables, probability mass function (PMF), cumulative distribution function (CDF), Bernoulli distribution, categorical distribution. | Chap. 4 |

8 | 21 March 2018 | Lecture 6 | Expectation and variance, algebraic rules for expectation, f/F notation for PMF and CDF, expectation and variance of Bernoulli distribution, demonstration of R statistical computing (simulation of data, mean, variance computation, plot of histogram and empirical CDF). | Chap. 4, parts of Chap. 7 |

Easter Break | ||||

9 | 17 April 2018 | Tutorial 3 | Example sheet 3 and Solutions 3. | |

9 | 17 April 2018 | Lecture 7 | Overview of discrete distributions, Binomial distribution (mean, variance, probability mass function), geometric distribution (mean, variance, probability mass function). | Chap. 4 |

9 | 18 April 2018 | Lecture 8 | Poisson distribution (mean, variance, probability mass function), relationship of Poisson and Binomial distribution, R demonstrations, credible intervals, statistics and probabilistic modeling, use of data to infer model parameters (method of moments). | Chap. 12 |

10 | 24 April 2018 | Tutorial 4 | Example sheet 4 and Solutions 4. | |

10 | 24 April 2018 | Lecture 9 | Pascal's triangle to compute Binomial coefficients, introduction to continuous random variables (probability density function vs. probability mass function), normal distribution and exponential distribution and their properties. | Chap. 5 |

10 | 25 April 2018 | Lecture 10 | Probability density function of normal distribution and exponetial distribution, R demonstrations to visualise changes in shape, scale and location, Standardisation of random variables, standard normal distribution, cumulative distribution function of standard normal random variables (Phi function), application to compute probabilities of interval events of normal random variables with given mean and variance. | Chap. 5 |

11 | 1 May 2018 | Coursework test (4pm) | ||

12 | 8 May 2018 | Tutorial 5 | Example sheet 5 and Solutions 5. | |

12 | 8 May 2018 | Lecture 11 | Revision | |

12 | 9 May 2018 | Lecture 12 | Revision | |

12 | 16 May 2018 | Exam (2pm) |