theory and problems of probability, random variables, and random processes - hwei p. hsu

[...]... intuitive notion of probability measure obtained from the notion of relative frequency C Elementary Properties of Probability: By using the above axioms, the following useful properties of probability can be obtained: 6 If A,, A , , .,A, are n arbitrary events in S, then - ( - 1 ) " - ' P ( A 1 n A, n - - n A,) (1.30) where the sum of the second term is over all distinct pairs of events, that of the third... (or disjointness) and independence of a collection of events we summarize as follows: 1 If (A,, i = 1,2, ,n} is a sequence of mutually exclusive events, then P( i)A,) i= 1 P(AJ = i= 1 2 If {A,, i = 1,2, ,n) is a sequence of independent events, then and a similar equality holds for any subcollection of the events Solved Problems SAMPLE SPACE AND EVENTS 1.1 Consider a random experiment of tossing a coin... n n A ,- ,) Multiplying both sides by P(A,+, I A , n A , n n A,), we have P(Al n A, n and - - - n A,)P(A,+,IA, n A, n A,) = P(Al n A , n P(A, n A , n - n A,, , ) = P(A,)P(A, 1 A,)P(A31 A , rl A,) Thus, Eq (1.81) is also true for n for n 2 2 1.45 n =k - n - - P(A,+, 1 A , A,,,) n A, n - n A,) + 1 By Eq ( 1 A l ) , Eq (1.81) is true for n = 2 Thus Eq (1.81) is true Two cards are drawn at random from... (1.l8) and (1.3), we have Taking complements of both sides of the above yields which is Eq (1 l9) 16 PROBABILITY [CHAP 1 THE NOTION AND AXIOMS OF PROBABILITY 1.18 Using the axioms of probability, prove Eq (1.25) We have S = A u A and AnA=@ Thus, by axioms 2 and 3, it follows that P(S) = 1 = P(A) + P(A) from which we obtain P(A) = 1 - P(A) 1.19 Verify Eq (1.26) From Eq (1Z), have we P(A) = 1 - P(A)... (1.25), P(B) = 1 - P(A) (b) Substituting n = 50 in Eq (1.78), we have P(A) z 0.03 and P(B) z 1 - 0.03 = 0.97 (c) From Eq (1.78), when n = 23, we have P(A) x 0.493 and P(B) = 1 - P(A) w 0.507 That is, if there are 23 persons in a room, the probability that at least two of them have the same birthday exceeds 0.5 1.33 A committee of 5 persons is to be selected randomly from a group of 5 men and 10 women (a)... r; 0) and i= 1 i= 1 1.11 Consider the switching networks shown in Fig 1-5 Let A,, A,, and A, denote the events that the switches s,, s,, and s, are closed, respectively Let A,, denote the event that there is a closed path between terminals a and b Express A,, in terms of A,, A,, and A, for each of the networks shown (4 (b) Fig 1-5 From Fig 1-5 (a), we see that there is a closed path between a and b... consists of 3 points (see Fig 1-3 ): (d) The event C is an impossible event, that is, C = 1 ( 2 CHAP 1) PROBABILITY A Fig 1-3 1.6 An automobile dealer offers vehicles with the following options: (a) With or without automatic transmission (b) With or without air-conditioning (c) With one of two choices of a stereo system (d) With one of three exterior colors , If the sample space consists of the set of all... Find the probability that the committee consists of 2 men and 3 women (b) Find the probability that the committee consists of all women (a) The number of total outcomes is given by It is assumed that "random selection" means that each of the outcomes is equally likely Let A be the event that the committee consists of 2 men and 3 women Then the number of outcomes belonging to A is given by Thus, by Eq... (1.2),A = @ = S, and by axiom 2 we obtain P(@)=l-P(S)=l-1=0 1.20 Verify Eq (1.27) Let A c B Then from the Venn diagram shown in Fig 1-7 , we see that B=Au(AnB) and An(AnB)=@ Hence, from axiom 3, P(B) = P(A) + P(A n B) However, by axiom 1, P(An B) 2 0 Thus, we conclude that P(A)IP(B) ifAcB Shaded region:A nB Fig 1-7 1.21 Verify Eq (1 29) From the Venn diagram of Fig 1-8 , each of the sets A u B and B can be... is a closed path between a and b if s, and s, or s, and s, are closed From Fig 1-5 (d), we see that there is a closed path between a and b if either s, and s, are closed or s, is closed Thus A,, = (A, n A,) u A3 PROBABILITY 1.12 [CHAP 1 Verify the distributive law (1.12) Let s E [ A n ( B u C)] Then s E A and s E (B u C) This means either that s E A and s E B or that s E A and s E C; that is, s E ( A . Schaum's Outline of Theory and Problems of Probability, Random Variables, and Random Processes Hwei P. Hsu, Ph.D. Professor of Electrical Engineering Fairleigh Dickinson University Start of. Professional[/PU][DP]1997[/DP]End of Citation Preface The purpose of this book is to provide an introduction to principles of probability, random variables, and random processes and their applications. The. Supervisor: Maureen Walker Library of Congress Cataloging-in-Publication Data Hsu, Hwei P. (Hwei Piao), date Schaum's outline of theory and problems of probability, random variables, and random