## MA200-5-AU-CO:

Statistics II

## Key module for

BSC N233DT Actuarial Science (Including Placement Year),

BSC N323 Actuarial Science,

BSC N323DF Actuarial Science,

BSC N323DT Actuarial Science,

BSC N324 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 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,

BSC N333 Actuarial Studies,

BSC N333DT Actuarial Studies,

BSC N334 Actuarial Studies (Including Placement Year),

BSC N334DT Actuarial Studies (Including Placement Year),

BSC N335 Actuarial Studies (Including Year Abroad)

## Module description

This module introduces distribution theory, estimation and Maximum Likelihood estimators, hypothesis testing ending by exploring basic linear regression and multiple linear regression implemented in R.

This module uses the R software environment for statistical computing and graphics.

## Module aims

The aims of this module are:

- To examine in detail several parametric distributions (Binomial, Poisson, Uniform, Normal, Exponential, Gamma, Chi-square, t, F) and explore their applications for modelling various real data settings.
- To explore the concept of multivariate distribution (e.g., copulas) in modelling the association between multiple variables, and understand the relations between joint, marginal distributions and independence.
- To learn the change of variables formula and know how to derive the distributions of functions of random variables.
- To understand the concepts of moment estimators and maximum likelihood estimators of unknown parameters. Be able to define the bias and mean squared error, and determine the efficiency of unbiased estimators using the Cramer-Rao theorem.
- To implement the linear regression model in studying the relationship between various variables, and determine the least squares estimates of the parameters. Conduct a diagnostic study of the model and its assumptions.
- To understand the concepts of hypothesis test, null and alternative hypotheses, type I and type II errors, test statistic, critical region, level of significance, probability-value and power of a test.
- To conduct Z-test, one-sample t-test, two-samples t-test, and F-test for evaluating hypotheses tests in real data situations.
- To understand the concept of interval estimation, and the use of central limit theorem in deriving confidence intervals.

## Module learning outcomes

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

- Study the properties of discrete distributions (Bernoulli, Binomial, Poisson) and continuous distributions (Uniform, Normal, Exponential, Gamma, Chi-square, t, F), and be familiar with their real applications.
- Handle bivariate distributions, and understand the relations between joint, marginal, conditional distributions and independence.
- Employ the change of variables formula in deriving the distributions of functions of random variables.
- Determine the moments estimators and maximum likelihood estimators of unknown parameters. Be able to define the bias and mean squared error of estimators, and determine the efficiency w.r.t. the Cramer-Rao lower bound for unbiased estimators.
- Implement the linear regression model in studying the relationship between various variables, and compute the least squares estimates of the parameters. Conduct a diagnostic study of the model and its assumptions.
- Understand the concepts of hypothesis test, null and alternative hypotheses, test statistic, critical region, type I and type II errors, level of significance, probability-value and power of a test.
- Conduct Z-test, one-sample t-test, two-samples t-test, and F-test for evaluating hypotheses tests in real data situations.
- Derive interval estimators of unknown parameters, and understand the relation between hypothesis tests and confidence intervals.
- Use R to implement the methods discussed in (1)-(8).

## Module information

The module follows the Graduate (Level 6) standards in Statistics of the Royal Statistical Society.

*Indiciative Syllabus:*

Distribution theory

- Sums of IID random variables, strong and weak law of large numbers, central limit theorem.
- Concept of Monte Carlo simulation.
- Joint, marginal and conditional distributions. Independence. Covariance and correlation.
- Convolution formula and change of variable formula.
- Moment generating functions to find moments of the PDF and distributions of sums of random variables.

Estimation

- Sampling distributions.
- Bias in estimators and mean squared error, efficiency and the Cramer-Rao lower bound for unbiased estimators.
- Maximum likelihood estimation and finding estimators analytically.
- The mean and variance of a sample mean.
- The distribution of the t-statistic for random samples from a normal distribution. - The F distribution for the ratio of two sample variances from independent samples taken from normal distributions.
- Chi Square distributions for the sum of squared standard normal variates

Hypothesis testing and Confidence intervals

- Confidence intervals for means, variances and differences between means.
- Hypothesis tests concerning means and variances.
- Null and alternative hypotheses, type I and type II errors, test statistic, critical region, level of significance, probability-value and power of a test.
- Use R to implement methods discussed above.

Linear models

- Linear relationships between variables using regression analysis.
- The correlation coefficient for bivariate data and the coefficient of determination. Response and explanatory variables and the least squares estimates of the slope and intercept parameters in a simple linear regression model.
- Multiple linear regression with IID normal errors, implemented in R.
- Use R to implement methods discussed above.

## Learning and teaching methods

Teaching in the School 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*

- Devore, J.L. and Berk, K.N.B. (2021)
*Modern Mathematical Statistics with Applications*. 3rd edn. Springer. Available at: https://ebookcentral.proquest.com/lib/universityofessex-ebooks/detail.action?docID=6578947. - Wackerly, D.D., Mendenhall, W. and Scheaffer, R.L. (2008)
*Mathematical statistics with applications*. Seventh edfition. Belmont, CA: Thomson Brooks/Cole. Available at: https://app.kortext.com/Shibboleth.sso/Login?entityID=https://idp0.essex.ac.uk/shibboleth&target=https://app.kortext.com/borrow/213483.

The library makes provision for all reading list items, with digital provision where possible, and these resources are shared between students.

Further reading can be obtained from this module's reading list.

## Assessment items, weightings and deadlines

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

Coursework | Test | ||

Exam | Main exam: In-Person, Open Book (Restricted), 180 minutes during Summer (Main Period) | ||

Exam | Reassessment Main exam: In-Person, Open Book (Restricted), 180 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

**35**hours,

**32 (91.4%)**hours available to students:

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

**2**hours not recorded due to opt-out by lecturer(s), module, or event type.

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

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