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MA12013: Core pure and statistical mathematics B

[Page last updated: 23 October 2023]

Academic Year: 2023/24
Owning Department/School: Department of Mathematical Sciences
Credits: 15 [equivalent to 30 CATS credits]
Notional Study Hours: 300
Level: Certificate (FHEQ level 4)
Period:
Semester 2
Assessment Summary: CWSI 12%, EXCB 88%
Assessment Detail:
  • Algebra 1b (EXCB 40%)
  • Probability & Statistics 1b (EXCB 28%)
  • Probability & Statistics 1b, Coursework 1 (CWSI 3%)
  • Probability & Statistics 1b, Coursework 2 (CWSI 3%)
  • Probability & Statistics 1b, Coursework 3 (CWSI 3%)
  • Probability & Statistics 1b, Coursework 4 (CWSI 3%)
  • Functions (EXCB 20%)
Supplementary Assessment:
Like-for-like reassessment (where allowed by programme regulations)
Requisites: Before taking this module you must take MA12012
Learning Outcomes: By the end of the unit, you will be able to:
  • Demonstrate understanding of linear algebra, including bases, dimension, rank, orthogonality, diagonalisation, including proofs. Perform computations with determinants, matrix inverses, orthonormal bases, eigenvalues, and eigenvectors.
  • Calculate properties of continuous random variables, manipulate joint probability distributions; formulate, fit and assess statistical models; use the R statistical package for simulation and data exploration.
  • Demonstrate fluency and familiarity with scalar-valued functions, including elementary definitions and results and applying these concepts to a range of examples.



Synopsis: You will develop further skills in mathematics. You will study linear algebra, both via matrices and via vector spaces and linear maps, focussing on the most widely applicable aspects. You will examine the theory of continuity for functions of one real variable. You will use continuous random variables and explore statistical modelling and parameter estimation.

Content: Algebra: Vector spaces (over the rational, real, and complex numbers), linear maps, subspaces, kernel, and image. Linear independence, spans, bases, and dimension. Matrix of a linear map, change of basis, linear operators, similar matrices. Rank, nullity and the Rank-Nullity theorem. Inner-product spaces: orthogonality, Cauchy-Schwarz inequality and applications. Gram-Schmidt process: orthonormal bases, orthogonal matrices and QR decomposition, orthogonal subspaces. Determinants and adjugates: properties and computation; invertibility of matrices. Eigenvalues, eigenvectors and eigenspaces; characteristic polynomial, algebraic and geometric multiplicities, diagonalisability. Self-adjoint operators, symmetric/Hermitian matrices and their orthogonal diagonalisation. Probability & Statistics: Properties of continuous random variables, common continuous distributions including the uniform, exponential, normal and gamma distributions. Transformations of random variables (RVs). Proof of the law of the unconscious statistician for a continuous RV. Joint probability density function (PDF). Marginal and conditional distributions of continuous RVs. Independence: factorisation of joint PDF as a product of marginals. Properties of covariance and correlation. Distribution of a sum of continuous RVs, including normal and exponential examples. Statement and application of the central limit theorem. Introduction to model fitting. Exploratory data analysis and model formulation. Parameter estimation by the method of moments and maximum likelihood. Estimators as random variables. Sampling distributions of estimators. Bias, variance and mean-square error of an estimator. Graphical assessment of goodness of fit. Functions: Limits. Continuity. Differentiability. Real power series

Course availability:

MA12013 is Compulsory on the following courses:

Department of Computer Science
  • USCM-AFB32 : BSc(Hons) Computer Science and Mathematics (Year 1)
  • USCM-AKB32 : BSc(Hons) Computer Science and Mathematics with professional placement (Year 1)
  • USCM-AKB32 : BSc(Hons) Computer Science and Mathematics with study abroad (Year 1)
  • USCM-AFM32 : MComp(Hons) Computer Science and Mathematics (Year 1)
  • USCM-AKM32 : MComp(Hons) Computer Science and Mathematics with professional placement (Year 1)
  • USCM-AKM32 : MComp(Hons) Computer Science and Mathematics with study abroad (Year 1)
Department of Economics
  • UHES-AFB12 : BSc(Hons) Economics and Mathematics (Year 1)
  • UHES-AKB12 : BSc(Hons) Economics and Mathematics with professional placement (Year 1)
  • UHES-AKB12 : BSc(Hons) Economics and Mathematics with professional placement and study abroad (Year 1)
  • UHES-AKB12 : BSc(Hons) Economics and Mathematics with study abroad (Year 1)

Notes:

  • This unit catalogue is applicable for the 2023/24 academic year only. Students continuing their studies into 2024/25 and beyond should not assume that this unit will be available in future years in the format displayed here for 2023/24.
  • Courses and units are subject to change in accordance with normal University procedures.
  • Availability of units will be subject to constraints such as staff availability, minimum and maximum group sizes, and timetabling factors as well as a student's ability to meet any pre-requisite rules.
  • Find out more about these and other important University terms and conditions here.