## Probability 1

 Unit code: MATH10141 Credit Rating: 10 Unit level: Level 1 Teaching period(s): Semester 1 Offered by School of Mathematics Available as a free choice unit?: N

None

#### Aims

The aims of this course are to introduce the basic ideas and techniques of probability, including the handling of random variables and standard probability distributions and the crucial notions of conditional probability and of independence.

#### Overview

The course gives a general introduction to probability and statistics and is a prerequisite for all future probability and statistics courses.

#### Learning outcomes

On successful completion of this module students will be able to:

• calculate various probabilities on a discrete probability space through counting arguments using combinations, permutations, factorials and multinomials;
• state, prove, apply, and extend various identities for the set operations in probability;
• calculate probabilities through appropriate application of conditional probabilities and independence;
• evaluate and prove properties of the expectation and variance of random variables in both the discrete and continuous setting including Bernoulli, Binomial, Geometric, Poisson, and Normal distributions;
• convert distributions between normal distributions and the standard normal distribution and state their application in results such as the the weak law of large numbers and central limit theorem;
• state, prove, apply and extend probabilistic inequalities such as Markov's inequality and Chebychev's inequality.

#### Assessment methods

• Other - 20%
• Written exam - 80%

#### Assessment Further Information

• Coursework; Weekly in class tests in the computer cluster, weighting within unit 20%
• Two hours end of semester examination; Weighting within unit 80%

#### Syllabus

1. Random experiments, sample space and events, the algebra of events (sets, unions, intersections, complementations, De Morgan’s laws). Axioms of probability. Equally likely events. Combinatorial probability. [4]
2. Conditional probability of an event. Multiplication rule. Partition theorem, Bayes' theorem and applications. Independent events. [4]
3. Random variables. Definition. Distribution function. Discrete random variables and probability mass function. Continuous random variables, probability density function and its relation to the distribution function. Calculating probabilities of events defined by random variables. Finding the distribution function of random variables using equivalent events (discrete functions only). [3]
4. Expectation and variance of a random variable and of a function of a random variable (including standardising). Basic properties of expectation and variance. [2]
5. Probability distributions including the Binomial, Geometric, Poisson, Normal and Exponential distributions. Standardisation of Normal variables. Poisson and Normal approximation to Binomial. [3]
6. Independent random variables. Expectation and variance of a linear combination of independent random variables. Discussion of the Normal case. [2]
7. Independent trials. Chebychev’s inequality. Weak Law of Large Numbers. The Central Limit Theorem. [4]

• S. Ross. A First Course in Probability, Macmillan.
• D. Stirzaker. Elementary Probability, Cambridge University Press. Available electronically
• J. McColl. Probability, London : Edward Arnold, 1995.
• N.A. Weiss, A Course in Probability, Pearson.

#### Feedback methods

Feedback seminars will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

#### Study hours

• Lectures - 22 hours
• Tutorials - 11 hours
• Independent study hours - 67 hours

#### Teaching staff

Neil Walton - Unit coordinator