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# Module Code - Title:

MS4222 - INTRODUCTION TO PROBABILITY AND STATISTICS

2022/3

2

0

1

0

7

6

N

# Rationale and Purpose of the Module:

This module replaces existing module MS4212 Introduction to Data Analysis. The focus of the previous module MS4212 was the analysis of data without a formal background in probability. The philosophy underpinning this approach was to introduce students to real data, which was entirely absent from Leaving Certificate mathematics in the 1990s, and begin to lay the foundations for the elements of data modelling necessary for the years three and four modules in the statistics options. Probability and Statistics account for 20% of the new Project Maths syllabus. Students now entering first year have had prior exposure to elementary data handling skills and experience applying the some basic ideas of probability. Consequently, it is not obvious that it is still necessary or desirable to adopt a teaching approach that separates the subject areas statistics and probability. As things stand, probability is totally absent from MS4212. One consequence of this omission is that statistical tools are introduced without proper formal theoretical justification based on probability models. Likewise, students are not as well prepared as they could be for the (rather packed) follow-on module MS4213. The intention in the revised (and renamed) first year introductory module is to include some probability in the syllabus. The strategy is to give students time to explore some of the many classical/famous problems that often arise in introductory probability. Discrete random variables and probability mass functions will be covered. As well as relieving some of the pressure in the congested semester 3 module MS4213, students will now be required to engage in more algebraic manipulation and basic mathematics. The statistical content of the module has been reconfigured to allow the inclusion of the material on probability.

# Syllabus:

Elementary Probability: permutations and combinations; axioms, rules of probability; conditional probability; independent events; probability trees; law of total probability; Bayes' rule. Discrete Random Variables: probability mass functions (Bernoulli, binomial, Poisson, geometric); expected value, variance; Poisson approximation to the binomial; law of total expectation (discrete form). The Normal Curve: the normal curve as an idealised histogram; areas under the normal curve; normal probability plot; illustrating the sampling distribution of the mean through applications in statistical quality control; precision of an estimate; the foundations of hypothesis testing and confidence intervals. Gathering Data: sample surveys; designed experiments and observational studies; randomized control trials. Exploratory Data Analysis: frequencies; histogram; empirical density curve; percentiles; measures of centre; measures of spread; outliers; boxplots; scatterplots; correlation; contingency tables, Simpson's Paradox. Regression Models: least squares line; transforming to linearity; out-of-sample prediction.

# Learning Outcomes:

## Cognitive (Knowledge, Understanding, Application, Analysis, Evaluation, Synthesis)

On successful completion of this module, students should be able to: Apply the rules of probability and conditional probability to applied real-world problems as well as classical problems in probability such as: collecting coupons; the prisoner's dilemma; the poly urn; the unsafe subway; the birthday problem(s). Calculate probabilities, expected values and variances, normalizing constants, of probability mass functions. Critique a designed experimental, clinical trial, or sample survey for sources of bias, confounding of effects, and apply randomization and blocking strategies. Use the statistical software package R to produce graphical and numerical summaries of data, applying transformations of variables to correct for skewness / non-linearity. Calculate probabilities based on the application of the Normal distribution and construct confidence intervals and test hypotheses about population means. Describe, quantify the strength of and model the relationship between two quantitative variables.

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# How the Module will be Taught and what will be the Learning Experiences of the Students:

Normal teaching delivery, including possible use of statistical software. Graduates will have a high level of competence in basic probability and statistics and a demonstrated capacity to bring statistical knowledge to bear on real world problems and challenges. Graduates can convey elementary concepts of probability and statistics clearly, effectively and professionally to a range of different stakeholders.

# Prime Texts:

F.M. Dekking, C. Kraaikamp, H.P. Lopuhaa, L.E. Meester (2005) A Modern Introduction to Probability and Statistics , Springer

# Other Relevant Texts:

David Moore , George P. McCabe , Bruce Craig (2012) Introduction to the Practice of Statistics , W.H. Freeman & Company

# Programme(s) in which this Module is Offered:

BAJOHOUFA - Joint Honours
BSFIMAUFA - Financial Mathematics
BSECMSUFA - Economics and Mathematical Sciences
BSSCCHUFA - Science Choice
BSMSCIUFA - Mathematical Sciences
BSMAPHUFA - Mathematics and Physics
BLLAPLUFA - Law Plus