Statistics II

The details

Requisites for this module

BSC N233 Actuarial Science (Including Placement Year),

BSC N323 Actuarial Science,

BSC N324 Actuarial Science (Including Year Abroad),

BSC N325 Actuarial Science (Including Foundation Year),

BSC C831 Cognitive Science,

BSC C832 Cognitive Science (Including Year Abroad),

BSC C833 Cognitive Science (Including Placement 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 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 GCF3 Mathematics with Physics (Including Year Abroad),

BSC I1G3 Data Science and Analytics,

BSC I1G3CE Data Science and Analytics,

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

BSC I1GBCE 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),

MSCIG199 Mathematics and Data Science

BSC N323 Actuarial Science,

BSC N324 Actuarial Science (Including Year Abroad),

BSC N325 Actuarial Science (Including Foundation Year),

BSC C831 Cognitive Science,

BSC C832 Cognitive Science (Including Year Abroad),

BSC C833 Cognitive Science (Including Placement 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 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 GCF3 Mathematics with Physics (Including Year Abroad),

BSC I1G3 Data Science and Analytics,

BSC I1G3CE Data Science and Analytics,

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

BSC I1GBCE 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),

MSCIG199 Mathematics and Data Science

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.

This module aims to cover part of the CS1 IFOA syllabus. The module follows the Graduate (Level 6) standards in Statistics of the Royal Statistical Society, see under http://www.rss.org.uk/Images/PDF/pro-dev/2016/rss-level6-standards.pdf

On completion of the module students should be able to:

(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.

(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.

MA108 is a prerequisite for this module. This module also requires basic mathematical knowledge of algebra, permutations, combinations and summation of series with notation. Competent knowledge of differential and integral calculus, including partial derivatives and double integrals, is certainly required.

Syllabus

Distribution theory

- Standard distributions and their use in modelling, including Bernoulli, binomial, Poisson, discrete uniform, Normal, exponential, continuous, uniform and multivariate Normal.

- Expectation, variance and generating functions.

- Sums of IID random variables, weak law of large numbers, central limit theorem.

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

- 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 tables of the t-, F-, and chi-squared distributions.

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.

Syllabus

Distribution theory

- Standard distributions and their use in modelling, including Bernoulli, binomial, Poisson, discrete uniform, Normal, exponential, continuous, uniform and multivariate Normal.

- Expectation, variance and generating functions.

- Sums of IID random variables, weak law of large numbers, central limit theorem.

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

- 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 tables of the t-, F-, and chi-squared distributions.

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.

Teaching will be delivered in a way that blends face-to-face classes, for those students that can be present on campus, with a range of online lectures, teaching, learning and collaborative support.

This module does not appear to have any essential texts. To see non-essential items, please refer to the module's reading list.

Coursework / exam | Description | Deadline | Weighting |
---|---|---|---|

Coursework | Test | ||

Exam | 180 minutes during Summer (Main Period) (Main) |

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

30% | 70% |

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

30% | 70% |

Module supervisor and teaching staff

Availability

Resources

Further information

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