Garantie de satisfaction à 100% Disponible immédiatement après paiement En ligne et en PDF Tu n'es attaché à rien
logo-home
F U N D A M E N T A L S OF PROBABILITY WITH STOCHASTIC PROCESSES SOLUTIONS TO SELF-QUIZZES AND SELF-TESTS 4th Edition. $15.74   Ajouter au panier

Examen

F U N D A M E N T A L S OF PROBABILITY WITH STOCHASTIC PROCESSES SOLUTIONS TO SELF-QUIZZES AND SELF-TESTS 4th Edition.

 14 vues  0 fois vendu
  • Cours
  • Fundamentals of Probability 4th Edition
  • Établissement
  • Fundamentals Of Probability 4th Edition

F U N D A M E N T A L S OF PROBABILITY WITH STOCHASTIC PROCESSES SOLUTIONS TO SELF-QUIZZES AND SELF-TESTS 4th Edition.

Aperçu 4 sur 108  pages

  • 27 septembre 2024
  • 108
  • 2024/2025
  • Examen
  • Questions et réponses
book image

Titre de l’ouvrage:

Auteur(s):

  • Édition:
  • ISBN:
  • Édition:
  • Fundamentals of Probability 4th Edition
  • Fundamentals of Probability 4th Edition
avatar-seller
Topscorer
SOLUTIONS TO SELF-QUIZZES AND SELF-TESTS

CHAPTER 1



FUNDAMENTALS
OF PROBABILITY
WITH STOCHASTIC PROCESSES




FOURTH EDITION




SAEED GHAHRAMANI
Western New England University
Springfield, Massachusetts, USA




A CHAPMAN & HALL BOOK

, Chapter 1 Solutions to Self-Quiz Problems 2



SOLUTIONS TO SELF-QUIZ PROBLEMS

Section 1.2

1. Suppose that jabk represents the order in which Jody is standing on the far left, Ann
next to Jody, Bill next to Ann, and Karl next to Bill. Define all other sample points
(possible outcomes) similarly. The event that, on the line, males and females alternate
is
E = {jbak, jkab, abjk, akjb, bjka, bakj, kjba, kabj}.
100

2. (a) E1E2 · · · E100. (b) E1 E2 · · · Ei−1 Eic Ei+1 · · · E100.
i=1

3.
(E ∪ F )(F ∪ G)(EG ∪ F c ) = (F ∪ E)(F ∪ G)(EG ∪ F c )
= (F ∪ EG)(EG ∪ F c )
= (EG ∪ F )(EG ∪ F c )
= EG ∪ F F c = EG ∪ ∅ = EG.

4. The event that a signal fed to the input is transmitted to the output is
F = E1 E2 E4 E7 ∪ E1 E2 E5 E6 E7 ∪ E1 E3 E5 E6 E7 ∪ E1 E3 E4 E7 .


Section 1.4

1. The desired probability is
 
P (Ac B c ) = 1−P A∪B = 1−P (A)−P (B)+P (AB) = 1−1/3−1/4+0 = 5/12.

2. Let A be the event that Zack reads the book assigned on the given period. Let B
be the event that he completes his philosophy paper on or before the deadline. The
probability that he ends up meeting the deadline only for one assignment is
 
P (A ∪ B) − AB = P (A ∪ B) − P (AB) = 0.95 − 0.6 = 0.35.
Note that this equality holds since AB ⊆ A ∪ B.

3. Let F , E, and I be the events that a randomly selected American traveler to Europe
travels to France, Englad, and Italy, respectively. We have
 
P F ∪ E ∪ I = P (F ) + P (E) + P (I) − P (F E) − P (F I) − P (EI) + P (EF I).
Therefore,
2 1 1 1 1 1 1
= + + − − − + P (EF I).
3 3 2 4 5 5 5
This gives that P (EF I) = 11/60.

, Chapter 1 Solutions to Self-Quiz Problems 3


Section 1.7

1. No, it does not! In the experiment of choosing a random point from the interval
(0, 1), let A be the event that the outcome is either 1/3 or 1/2, and B be the event that
the outcome is either 1/5 or 1/3. Then
 AB = {1/3}. So A and B are not mutually
exclusive. However, P (AB) = P {1/3} = 0.

2. For the experiment of choosing a point at random from the interval [0, 1], let E n =
1 1 1 2 
− , + , n ≥ 1. Applying the Continuity of Probability Function
3 n+2 3 n+2
to
En ’s, show that P (1/3 is selected) = 0.

, Chapter 1 Solutions to Self-Test Problems 4




SOLUTIONS TO SELF-TEST PROBLEMS


1. (a) By symmetry, the probability that Crispin ends up paying for dinner is 1/3.

(b) The sample space for one round of flips is

S = HHT, HTH, HTT, HHH, THT, TTH, THH, TTT .

The probability that no more than one round of flips is necessary is equal to the
probability that HHH and TTT do not occur in the first round, which is 6/8 = 3/4.

2. Let x1, x2, x3, x4 , and x5 be the production level, in hundreds, for baseballs, tennis
balls, softballs, basketballs, and soccer balls, respectively. A sample space for the
production levels of these balls by this manufacturer is

S = (x1 , x2 , . . . , x5) : 8 ≤ xi ≤ 13, 1 ≤ i ≤ 5 .

The event that on this day, the difference between the number of baseballs and soccer
balls produced will exceed 100 is

E = (x1 , x2, . . . , x5 ) ∈ S : |x1 − x5 | ≤ 1
= (x1 , x2, . . . , x5 ) ∈ S : x5 − 1 ≤ x1 ≤ x5 + 1 .

3. (a) The sample space is

S = {A1 A2 A3 , Ac1 A2 A3 , A1 Ac2 A3 , A1 A2 Ac3 , Ac1 Ac2 A3 , Ac1 A2 Ac3 , A1 Ac2 Ac3 , Ac1 Ac2 Ac3 }.

(b) The event that the system is not operative at the random time is

E = {Ac1 A2 A3 , A1 Ac2 A3 , A1 A2 Ac3 , Ac1 Ac2 A3 , Ac1 A2 Ac3 , A1 Ac2 Ac3 , Ac1 Ac2 Ac3 }.

4. Let A be the event that a randomly selected person from the given age group at this
geographic area suffers from Parkinson’s disease. Let B be the even that he or she
suffers from Alzheimer’s disease. The desired probability is
 
P (Ac B c ) = 1 − P A ∪ B = 1 − P (A) + P (B) − P (AB)
= 1 − (0.134 + 0.113 − 0.0226) = 0.7756.

This shows that 77.56% of the people in the given age group in the given geographic
area have neither of the two diseases.
 
5. We have that E cF = E ∪ F − E and E ⊆ E ∪ F . So
   
P E cF = P E ∪F −P (E) = 1−P (E c F c ) −P (E) = (1−0.35)−0.4 = 0.25.
 c
Note that by DeMorgan’s first law, E ∪ F = E cF c .

Les avantages d'acheter des résumés chez Stuvia:

Qualité garantie par les avis des clients

Qualité garantie par les avis des clients

Les clients de Stuvia ont évalués plus de 700 000 résumés. C'est comme ça que vous savez que vous achetez les meilleurs documents.

L’achat facile et rapide

L’achat facile et rapide

Vous pouvez payer rapidement avec iDeal, carte de crédit ou Stuvia-crédit pour les résumés. Il n'y a pas d'adhésion nécessaire.

Focus sur l’essentiel

Focus sur l’essentiel

Vos camarades écrivent eux-mêmes les notes d’étude, c’est pourquoi les documents sont toujours fiables et à jour. Cela garantit que vous arrivez rapidement au coeur du matériel.

Foire aux questions

Qu'est-ce que j'obtiens en achetant ce document ?

Vous obtenez un PDF, disponible immédiatement après votre achat. Le document acheté est accessible à tout moment, n'importe où et indéfiniment via votre profil.

Garantie de remboursement : comment ça marche ?

Notre garantie de satisfaction garantit que vous trouverez toujours un document d'étude qui vous convient. Vous remplissez un formulaire et notre équipe du service client s'occupe du reste.

Auprès de qui est-ce que j'achète ce résumé ?

Stuvia est une place de marché. Alors, vous n'achetez donc pas ce document chez nous, mais auprès du vendeur Topscorer. Stuvia facilite les paiements au vendeur.

Est-ce que j'aurai un abonnement?

Non, vous n'achetez ce résumé que pour $15.74. Vous n'êtes lié à rien après votre achat.

Peut-on faire confiance à Stuvia ?

4.6 étoiles sur Google & Trustpilot (+1000 avis)

80796 résumés ont été vendus ces 30 derniers jours

Fondée en 2010, la référence pour acheter des résumés depuis déjà 14 ans

Commencez à vendre!
$15.74
  • (0)
  Ajouter