## MA108-4-SP-CO:

Statistics I

## Key module for

BSC N233DT Actuarial Science (Including Placement Year),

BSC N323 Actuarial Science,

BSC N323DT Actuarial Science,

BSC N324 Actuarial Science (Including Year Abroad),

BSC N324DT Actuarial Science (Including Year Abroad),

BSC N325 Actuarial Science (Including Foundation Year),

BSC L1G2 Economics and Mathematics (Including Placement Year),

BSC LG11 Economics and Mathematics,

BSC LG18 Economics and Mathematics (Including Foundation Year),

BSC LG1C Economics and Mathematics (Including Year Abroad),

BSC L1G1 Economics with Mathematics,

BSC L1G3 Economics with Mathematics (Including Placement Year),

BSC L1G8 Economics with Mathematics (Including Foundation Year),

BSC L1GC Economics with Mathematics (Including Year Abroad),

BSC GN13 Finance and Mathematics,

BSC GN15 Finance and Mathematics (Including Placement Year),

BSC GN18 Finance and Mathematics (Including Foundation Year),

BSC GN1H Finance and Mathematics (Including Year Abroad),

BSC G100 Mathematics,

BSC G102 Mathematics (Including Year Abroad),

BSC G103 Mathematics (Including Placement Year),

BSC G104 Mathematics (Including Foundation Year),

MMATG198 Mathematics,

BSC 5B43 Statistics (Including Year Abroad),

BSC 9K12 Statistics,

BSC 9K13 Statistics (Including Placement Year),

BSC 9K18 Statistics (Including Foundation Year),

BSC G1G4 Mathematics with Computing (Including Year Abroad),

BSC G1G8 Mathematics with Computing (Including Foundation Year),

BSC G1GK Mathematics with Computing,

BSC G1IK Mathematics with Computing (Including Placement Year),

BSC G1F3 Mathematics with Physics,

BSC G1F4 Mathematics with Physics (Including Placement Year),

BSC G1F5 Mathematics with Physics (Including Foundation Year),

BSC GCF3 Mathematics with Physics (Including Year Abroad),

BSC I1G3 Data Science and Analytics,

BSC I1GB Data Science and Analytics (Including Placement Year),

BSC I1GC Data Science and Analytics (Including Year Abroad),

BSC I1GF Data Science and Analytics (Including Foundation Year),

MSCIN399 Actuarial Science and Data Science,

MSCIG199 Mathematics and Data Science

## Module description

This module introduces students to the basic ideas of probability (combinatorial analysis and axioms of probability), conditional probability and independence, probability distributions, and provides an introduction to handling data using descriptive statistics.

This module uses the R software package to clarify and illustrate theoretical concepts of probability, to show how random variables are generated and how they vary, enabling the construction of appropriate diagrams for data summary.

## Module aims

The aims of this module are:

- Familiarize students with descriptive statistics to explore real data.
- Understand the concepts of probability and conditional probability and apply them in real situations using various counting techniques.
- Understand the concepts of discrete and continuous random variables, their distributions, and their measures of location and variability.
- Examine in detail several parametric models (Binomial, Poisson, Normal, Gamma, etc) and explore their applications.
- Introduce students to R software and investigate data sets.

## Module learning outcomes

By the end of the module, students will be expected to:

- understand how to calculate and interpret simple summary statistics;
- how to choose and construct appropriate diagrams to illustrate data sets;
- use R for the data analysis examples of the course;
- understand and apply the addition rule and multiplication rule of probability;
- understand the basic ideas of conditional probability including the application of the total probability theorem and Bayes' theorem;
- understand and recognise situations appropriate for Binomial and Poisson models;
- calculate expectations and variances for discrete and continuous random variables, understand the ideas of probability density function and the distribution function;
- understand the change of variables formula and know how to calculate the distributions of functions of random variables;
- understand moment generating functions;
- recognise the central role of the normal distribution, be able to reduce normal random variables to standard form and be able to use tables of normal probabilities;
- understand the basic ideas of central limit theorem;
- implement in R the statistical methods described above.

## Module information

For students outside of the Department, an appropriate A level in Mathematics (or equivalent) is required for this module. If you are unsure whether you meet this criteria please contact maths@essex.ac.uk before attempting to enrol.

This module covers part of the CS1 IFOA syllabus.

*Indicative syllabus*

Descriptive statistics:

Data collection and summary

Stem/leaf plots and histograms

Measures of location (mode, median, mean)

Measures of spread

Quartiles

Box plots

Variance and standard deviation

Transformations

Probability:

Relative frequency

Probability as a limit

Events

Union and intersection

Addition rule

Exclusive events

Independent events

Multiplication rule

Permutations and combinations

Conditional probability

Total probability theorem

Bayes' theorem

Discrete probability distributions:

Discrete random variables

Probability distributions

Expectation

Algebra of expectations

Variance

Bernoulli distribution

Binomial distribution (sampling with replacement)

Mean and variance of Bernoulli and binomial

Poisson distribution (and applications)

Derivation of the Poisson

Approximation to the binomial

Continuous probability distributions:

Density function as limit of histograms

Properties of probability density function (pdf)

Cumulative distribution function (cdf)

Uniform and exponential distribution

Expectations

Variance

Median, mode

Distributions of functions of random variables

Change of variables formula

Normal distribution

Use of tables

Central limit theorem

Additivity

Moment generating function

Implement in R the statistical methods described above.

## Learning and teaching methods

Teaching in the department will be delivered using a range of face to face lectures, classes and lab sessions as appropriate for each module. Modules may also include online only sessions where it is advantageous, for example for pedagogical reasons, to do so.

## Bibliography*

## Assessment items, weightings and deadlines

Coursework / exam | Description | Deadline | Coursework weighting |
---|---|---|---|

Coursework | Test | ||

Exam | Main exam: In-Person, Open Book (Restricted), 90 minutes during Summer (Main Period) | ||

Exam | Reassessment Main exam: In-Person, Open Book (Restricted), 90 minutes during September (Reassessment Period) |

### Exam format definitions

**Remote, open book:**Your exam will take place remotely via an online learning platform. You may refer to any physical or electronic materials during the exam.**In-person, open book:**Your exam will take place on campus under invigilation. You may refer to any physical materials such as paper study notes or a textbook during the exam. Electronic devices may not be used in the exam.**In-person, open book (restricted):**The exam will take place on campus under invigilation. You may refer only to specific physical materials such as a named textbook during the exam. Permitted materials will be specified by your department. Electronic devices may not be used in the exam.**In-person, closed book:**The exam will take place on campus under invigilation. You may not refer to any physical materials or electronic devices during the exam. There may be times when a paper dictionary, for example, may be permitted in an otherwise closed book exam. Any exceptions will be specified by your department.

Your department will provide further guidance before your exams.

### Overall assessment

Coursework | Exam |
---|---|

30% | 70% |

### Reassessment

Coursework | Exam |
---|---|

30% | 70% |

## External examiner

**49**hours,

**33 (67.3%)**hours available to students:

**16**hours not recorded due to service coverage or fault;

**0**hours not recorded due to opt-out by lecturer(s).

*** Please note:** due to differing publication schedules, items marked with an asterisk (*) base their information upon the previous academic year.

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