# Module Overview

### Calculus and Statistics

The purpose of this module is to provide the learner with the statistical concepts and tools necessary for any engineering or science graduate, as well as underpinning specific engineering topics such as statistical process control, quality control and reliability analysis. To do this, the learner will cover the fundamental ideas of probability and descriptive statistics, moving on to Hypothesis testing and the design of experiments. They will then expand their knowledge of calculus to include more methods of integration and to the solution of number of classes of ordinary differential equations, including their treatment with transforms.

PIAA H2011

##### ECTS Credits

10

*Curricular information is subject to change

Power Series

Arithmetic and geometric sequences and series, their properties and applications. The concept of a McLaurin series and the method of finding the series for a range of common functions directly by repeated differentiation. The definition of McLaurin and Taylor series and their tests for convergence. The link with even and odd functions. Basic tests for convergence for series, including power series.

Ordinary Differential equations

The Methods of solution of Ordinary Differential Equations; first order linear equations and the integrating factor, separable first order equations, second order differential equations with constant coefficients and the characteristic equation. The concept of an engineering system and its modelling as a differential equation. The treatment of linear Ordinary Differential equations as systems solvable by the use of Phasors. Producing empirical approximate solutions for ordinary differential equations using the McLaurin series.

Transforms

The definition of the Laplace transform. The time domain and the s-domain. The transforms of common functions. Expanding the range of functions for which a transform can be found using the properties of transforms; the transform of the derivative of a function and the application of transforms to solving ordinary differential equations.

Descriptive Statistics

Calculation of measures of central tendency and dispersal: mean, mode, median and standard deviation. Grouped data; calculation of mean, mode, median and standard deviation for grouped data. Data representation and types of charts. Linear regression and correlation as data analysis techniques.

Probability

The definition of the fundamental ideas of events, experiments and probability. Independent events, conditional probabilities and the addition and multiplication laws. Permutations and combinations. The concepts of a random variable and its distribution, the definition of population parameters in terms of the probability distribution function and the cumulative probability distribution. Discrete and continuous probability distributions, including the exponential, normal, binomial and Poisson distributions. Examples of the role of these distributions in reliability prediction, component failures and designing for reliability.

Hypothesis testing

The concept of a statistic. The Central Limit Theorem and the concept of standard error. The common population parameters as statistics. The concept and limitations of a Hypothesis test, including type I and II errors and their probabilities. The representation of the results of a test; critical values and confidence intervals. Distributions including the ‘Student t’, the chi-square and the F distributions. Standard tests, including tests on means and variances, paired sample and unpaired tests on comparisons of means. Categorical tests using the chi-square distribution, such as goodness-of-fit tests to a distribution and tests for independence. Linear regression and correlation as statistical tests.

Design of Experiments and ANOVA

The design of experiments. Standard designs such as one way layout, block design and two way layout. The F distribution. The comparison of group means by one- and two-way analysis of variance.

Module Content & Assessment
Assessment Breakdown %
Formal Examination50
Other Assessment(s)50