## 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

## Module aims

## Module learning outcomes

(1) define and be familiar with the discrete distributions: binomial, Poisson and uniform and be familiar with the continuous distributions: normal, exponential, chi-square, t, F and uniform;

(2) use the one-to-one correspondence between an mgf and a pdf for sums of RVs;

(3) handle bivariate distributions, understanding the relations between joint, marginal, conditional distributions and independence;

(4) understand the uses of the central limit theorem;

(5) determine maximum likelihood and least squares estimates of unknown

parameters. Be able to define the terms: bias and mean squared error. Determine efficiency w.r.t. the Cramer-Rao lower bound for unbiased estimators;

(6) determine confidence intervals for means, variances and differences

between means;

(7) concepts of random sampling, statistical inference and sampling distribution, Hypothesis tests. 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 tables of the t-, F-, and chi-squared distributions;

(8) investigate linear relationships between variables using regression analysis. Use the correlation coefficient for bivariate data and the coefficient of determination. Explain what is meant by response and explanatory variables. Derive and calculate the least squares estimates of the slope and intercept parameters in a simple linear regression model. Perform multiple linear regression using R and interpret output;

(9) use R to implement the methods discussed in (1)-(8), [R Core Team (2017), R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, https://www.R-project.org/] for the data analysis examples of the module.

Apply R to do GLM and Monte-Carlo Simulation.

## Module information

Distribution theory

1. Sums of IID random variables, strong and weak law of large numbers, central limit theorem.

2. Concept of Monte Carlo simulation.

3. Joint, marginal and conditional distributions. Independence. Covariance and correlation.

4. Convolution formula and change of variable formula.

5. Moment generating functions to find moments of the PDF and distributions of sums of random variables.

Estimation

1. Sampling distributions.

2. Bias in estimators and mean squared error, efficiency and the Cramer-Rao lower bound for unbiased estimators.

3. Maximum likelihood estimation and finding estimators analytically.

4. The mean and variance of a sample mean.

5. 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.

6. Chi Square distributions for the sum of squared standard normal variates

Hypothesis testing and Confidence intervals

1. Confidence intervals for means, variances and differences between means.

2. 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.

Linear models

1. Linear relationships between variables using regression analysis.

2. 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.

3. Multiple linear regression with IID normal errors, implemented in R.

4. Use R to implement methods discussed above.

## Learning and teaching methods

## 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

**40**hours,

**30 (75%)**hours available to students:

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

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

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