Applied Statistics and Probability for Engineers, 6th Edition

Published by Wiley
ISBN 10: 1118539710
ISBN 13: 978-1-11853-971-2

Chapter 2 - Section 2-1 - Sample Spaces and Events - Exercises - Page 27: 2-23

Answer

(a) S={(d,d,d,d),(d,d,d,n),(d,d,n,d),(d,d,n,n),(d,n,d,d),(d,n,d,n),(d,n,n,d),(d,n,n,n),(n,d,d,d),(n,d,d,n),(n,d,n,d),(n,d,n,n),(n,n,d,d),(n,n,d,n),(n,n,n,d),(n,n,n,n)} (b) No, because $A_{1} \cap A_{2} \cap A_{3} \cap A_{4} \neq \varphi$ (c) $A_{1}$ ={(d,d,d,d),(d,d,d,n),(d,d,n,d),(d,d,n,n),(d,n,d,d),(d,n,d,n),(d,n,n,d),(d,n,n,n)} (d) $A_{1}^{'}$={(n,d,d,d),(n,d,d,n),(n,d,n,d),(n,d,n,n),(n,n,d,d),(n,n,d,n),(n,n,n,d),(n,n,n,n)} (e) $A_{1} \cap A_{2} \cap A_{3} \cap A_{4} $={(d,d,d,d)} (f) $\left(A_{1} \cap A_{2}\right) \cup\left(A_{3} \cap A_{4}\right)=\{(d,d,d,d), (d,n,d,d), (d,d,d,n),(n,d,d,d),(d,d,n,d), \text {(n,n,d,d),(d,d,n,n)}\}$

Work Step by Step

Let "d" denote a distorted bit, Let "n" denote a not distorted bit. (a) S={(d,d,d,d),(d,d,d,n),(d,d,n,d),(d,d,n,n),(d,n,d,d),(d,n,d,n),(d,n,n,d),(d,n,n,n),(n,d,d,d),(n,d,d,n),(n,d,n,d),(n,d,n,n),(n,n,d,d),(n,n,d,n),(n,n,n,d),(n,n,n,n)} Note that, $A_{1}$={(d,d,d,d),(d,d,d,n),(d,d,n,d),(d,d,n,n),(d,n,d,d),(d,n,d,n),(d,n,n,d),(d,n,n,n)} $A_{2}$={(d,d,d,d),(d,d,d,n),(d,d,n,d),(d,d,n,n),(n,d,d,d),(n,d,d,n),(n,d,n,d),(n,d,n,n)} $A_{3}$={(d,d,d,d),(d,d,d,n),(d,n,d,d),(d,n,d,n),(n,d,d,d),(n,d,d,n),(n,n,d,d),(n,n,d,n)} $A_{4}$={(d,d,d,d),(d,d,n,d),(d,n,d,d),(d,n,n,d),(n,d,d,d),(n,d,n,d),(n,n,d,d),(n,n,n,d)} (b) No, because $A_{1} \cap A_{2} \cap A_{3} \cap A_{4} \neq \varphi$ (c) $A_{1}$ ={(d,d,d,d),(d,d,d,n),(d,d,n,d),(d,d,n,n),(d,n,d,d),(d,n,d,n),(d,n,n,d),(d,n,n,n)} (d) $A_{1}^{'}$={(n,d,d,d),(n,d,d,n),(n,d,n,d),(n,d,n,n),(n,n,d,d),(n,n,d,n),(n,n,n,d),(n,n,n,n)} (e) $A_{1} \cap A_{2} \cap A_{3} \cap A_{4} $={(d,d,d,d)} (f) $\left(A_{1} \cap A_{2}\right) \cup\left(A_{3} \cap A_{4}\right)=\{(d,d,d,d), (d,n,d,d), (d,d,d,n),(n,d,d,d),(d,d,n,d), \text {(n,n,d,d),(d,d,n,n)}\}$
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