CambridgeInternational AS A LevelFurtherMathematicsFurther ProbabilityStatisticsJohn du FeuSeries editor:Roger PorkessDDYNAMICHODDEREDUCATIONAN HACHETTE UK COMPANYQuestions from the Cambridge International AS A Level Further Mathematics papers are reproducedby permission of Cambridge Assessment International Education.Unless otherwise acknowledged,thequestions,example answers,and comments that appear in this book were written by the authors.Cambridge Assessment International Education bears no responsibility for the example answers toquestions taken from its past question papers which are contained in this publication.IGCSE is a registered trademark.The publishers would like to thank the following who have given permission to reproduce photographsin this book:Photo creditsp.1 jakapan Chumchuen/Shutterstock;p.10 lcswart/Shutterstock;p.28 Barsan ATTILA/Shutterstock;p.78 Dean Drobot/Shutterstock;p.121Arthur Simoes/Shutterstock;p.156Vibeke Morfield/123RF.Every effort has been made to trace copyright and acknowledge ownership.The publishers will be gladto make suitable arrangements with any copyright holders whom it has not been possible to contact.Hachette UK's policy is to use papers that are natural,renewable and recyclable products and madefrom wood grown in sustainable forests.The logging and manufacturing processes are expected toconform to the environmental regulations of the country of origin.Orders:please contact Bookpoint Ltd,130 Park Drive,Milton Park,Abingdon,Oxon OX14 4SE.Telephone:(44)01235 827720.Fax:(44)01235 400401.Email:education@bookpoint.co.uk.Lines areopen from 9 a.m.to 5 p.m.,Monday to Saturday,with a 24-hour message answering service.You canalso order though our website:www.hoddereducation.comMuch of the material in this book was published originally as part of the MEI Structured Mathematicsseries.It has been carefully adapted for the Cambridge International AS A Level Further Mathematicssyllabus.The original MEI author team for Statistics comprised Alec Cryer,Michael Davies,Anthony Eccles,Bob Francis,Gerald Goodall,Alan Graham,Nigel Green,Liam Hennessy,Roger Porkessand Charlie Stripp.Roger Porkess and John du Feu 2018First published in 2018 byHodder Education,an Hachette UK company,Carmelite House,50 Victoria Embankment,London EC4Y ODZImpression number 5 4 3 2 1Year20222021202020192018All rights reserved.Apart from any use permitted under UK copyright law,no part of this publicationmay be reproduced or transmitted in any form or by any means,electronic or mechanical,includingphotocopying and recording,or held within any information storage and retrieval system,withoutpermission in writing from the publisher or under Licence from the Copyright Licensing Agency Limited.Further details of such licences(for reprographic reproduction)may be obtained from the CopyrightLicensing Agency Limited,www.cla.co.uk.Cover photo by Shutterstock/Ed SamuelIllustrations by Tech-Set,Aptara,Inc.,and Integra Software ServicesTypeset in Bembo Std 11/13 Integra Software Services Pvt Ltd,Pondicherry,IndiaPrinted in ItalyA cataloque record for this title is available from the British Library.ISBN9781510421813MIXPaper fromresponsible sourcesFSCwww.fsc.orgFSCC104740ContentsIntroductionivHow to use this bookThe Cambridge International AS A Level FurtherMathematics 9231 syllabus1 Continuous random variables11.1 Piecewise definition of a probability density function21.2 The expectation and variance of a function of X1.3 The cumulative distribution function101.4 Finding the PDF of a function of a continuous random variable192 Inference using normal and t-distributions282.1 Interpreting sample data using the t-distribution282.2 Using the t-distribution for paired samples342.3Hypothesis testing on a sample mean using the t-distribution352.4Using the t-distribution with two samples432.5Comparison between paired and 2-sample t-tests502.6Testing for a non-zero value of the difference of two means512.7Hypothesis tests and confidence intervals532.8Using the normal distribution with two samples602.9 Tests with large samples653 Chi-squared tests783.1 The chi-squared test for a contingency table793.2Goodness of fit tests934 Non-parametric tests1214.1 Single-sample non-parametric tests1214.2 Paired-sample non-parametric tests1334.3 The Wilcoxon rank-sum test1425 Probability generating functions1565.1 Probabilities defined by a probability generating function1575.2 Expectation and variance1615.3 The sum of independent random variables1675.4 The PGFs for some standard discrete probability distributions172Index179