Introductory description

FP006-30 Statistics and Further Mathematics

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Statistics

Numerical measures

• Standard deviation and variance calculated on ungrouped and grouped data
• Choice of numerical measures; mean, median, mode, range and interquartile range

Correlation and regression

• Calculation and interpretation of the product moment correlation coefficient
• Identification of response (dependent) and explanatory (independent) variables in regression
• Calculation of least squares regression lines with one explanatory variable. Scatter diagrams and drawing a regression line theorem
• Calculation of residuals

Probability

• Addition law of probability, mutually exclusive events
• Multiplication law of probability and conditional probability, independent events.

Discrete Random Variables

• Discrete random variables and their associated probability distributions
• Mean, variance and standard deviation
• Mean, variance and standard deviation of a simple function of a discrete random variable

Binomial Distribution

• Calculation of probability using formula
• Use of tables
• Mean, variance and standard deviation of a binomial distribution

Geometric Distribution

• Calculation of probability using formula
• Mean, variance and standard deviation of a Geometric distribution

Poisson Distribution

• Calculation of probability using formula
• Use of tables
• Mean, variance and standard deviation of a Poisson distribution
• Use the Poisson distribution to approximate binomial distributions

Continuous Random Variables

• Differences from discrete random variables
• Probability density functions, cumulative distribution functions and their relationship
• The probability of an observation lying in a specified interval
• Quartiles and percentiles
• Mean, variance and standard deviation
• Mean, variance and standard deviation of a simple function of continuous random variable

Continuous Uniform (Rectangular) Distribution

• General probability density function and cumulative distribution function
• Mean, variance and standard deviation

Normal Distribution

• Continuous random variables
• Properties of normal distributions
• Calculation of probabilities
• Mean, variance and standard deviation of a normal distribution
• Use the normal distribution to approximate binomial and Poisson distributions

Further Mathematics

Matrices

• Addition, subtraction and multiplication, inverse (2x2 and 3x3).
• Solving linear systems using inverse matrix method.
• Determinants (2x2 and 3x3).
• Solutions of simultaneous equations using Cramer's rule.
• Linear transformations.
• Matrix algebra, identity matrices.
• Reversing transformations using inverse matrix method.
• Calculating area enlargement scale factors using determinants.

Vectors

• Vector equations of lines.
• Intersection of lines
• Distance between lines: parallel and skew
• Dot product.
• Vector product. Application to area of triangle and parallelogram in 3d.
• Scalar triple product. Application to volume of parallelepiped and tetrahedron
• Vector equation of the plane.
• Intersection of a line and a plane
• Intersection of two planes

Partial Differentiation:

• Functions of several variables.
• Partial derivatives of functions of several variables.
• Directional derivatives.
• Total derivatives.
• Higher partial derivatives.
• Maxima, minima and saddle points.
• Lagrange multiplier method.
• Unconstrained optimization in economics.
• Constrained optimization in economics.

First Order Differential Equations

• Solving homogeneous first order differential equations.
• Solving linear first order differential equations using an integrating factor.
• Solving exact and non-exact first order differential equations.

Second Order Differential Equations

• Solutions of homogeneous second order differential equations.
• Solution of non-homogeneous second order differential equations by particular integrals.

Maclaurin and Taylor Series

• Maclaurin series. Standard series expansions.
• Taylor series expansions.
• Series solution to a differential equations.

Learning outcomes

By the end of the module, students should be able to:

• Analyse problems and use mathematical and statistical knowledge to plan strategies for solving them;
• Select and apply a range of mathematical and statistical tools and techniques that are appropriate to solve a given problem;
• Create and use appropriate mathematical and statistical models to solve and understand practical real-life problems, in appropriate contexts;
• Construct and present arguments through appropriate use of logical deduction and precise statements involving correct use of symbols and appropriate statistical and further mathematical language;
• Evaluate the outcomes and predictions of a mathematical or statistical model by considering any assumptions made, and its validity and limitations.

Indicative reading list

Statistics

Crawshaw, D.J. and Chambers, J.S. (2001) A Concise Course in Advanced Level Statistics (4th ed.) Nelson Thornes

Mathematics

C. M. McGregor; J. J. C. Nimmo; W. W. Stothers (2010) Fundamentals of university mathematics, (3rd ed.) Woodhead.

Gaulter, B. and Gaulter, M. (2001) Further pure mathematics. Oxford University Press.

View reading list on Talis Aspire

Subject specific skills

Mathematical Skills

Analytical Skills

Problem-solving skills

Transferable skills

Mathematical Skills

Analytical Skills

Problem-solving skills

Communication Skills