Solution Manual for Fundamentals of Probability with Stochastic Processes 4th Edition by Saeed Ghahramani 2019 PDF | Complete Step-by-Step Solutions and Worked Examples | Covers Probability Theory, Random Variables, Distributions, Expectation, Conditional

Covers All 12 Chapters




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C ontents

► 1 Axioms of Probability 1

1.2 Sample Space and Events 1
1.4 Basic Theorems 4
1.7 Random Selection of Points from Intervals 10
Review Problems 13


► Companion for Chapter 1 19

1B Applications of Probability to Genetics 19


► 2 Combinatorial Methods 21
2.2 Counting Principle 21
2.3 Permutations 24
2.4 Combinations 27
2.5 Stirling’ Formula 42
Review Problems 42


► 3 Conditional Probability and Independence 47

3.1 Conditional Probability 47
3.2 The Multiplication Rule 52
3.3 Law of Total Probability 55
3.4 Bayes’ Formula 60
3.5 Independence 66
Review Problems 76


► Companion for Chapter 3 80

3B More on Applications of Probability to Genetics 80


► 4
Distribution Functions and
Discrete Random Variables
85

4.2 Distribution Functions 85

,4.3 Discrete Random Variables 89
4.4 Expectations of Discrete Random Variables 94
4.5 Variances and Moments of Discrete Random Variables 100
4.6 Standardized Random Variables 107
Review Problems 107


► 5 Special Discrete Distributions 110

5.1 Bernoulli and Binomial Random Variables 110
5.2 Poisson Random Variable 117
5.3 Other Discrete Random Variables 125
Review Problems 132


► 6 Continuous Random Variables 137
6.1 Probability Density Functions 137
6.2 Density Function of a Function of a Random Variable 141
6.3 Expectations and Variances 144
Review Problems 151


► 7 Special Continuous Distributions 153

7.1 Uniform Random Variable 153
7.2 Normal Random Variable 158
7.3 Exponential Random Variables 166
7.4 Gamma Distribution 171
7.5 Beta Distribution 175
7.6 Survival Analysis and Hazard Function 180
Review Problems 183

► 8 Bivariate Distributions 187

8.1 Joint Distribution of Two Random Variables 187
8.2 Independent Random Variables 200
8.3 Conditional Distributions 209
8.4 Transformations of Two Random Variables 218
Review Problems 230

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► 9 Multivariate Distributions 238

9.1 Joint Distribution of n > 2 Random Variables 238
9.2 Order Statistics 248
9.3 Multinomial Distributions 253
Review Problems 255

► 10 More Expectations and Variances 260

10.1 Expected Values of Sums of Random Variables 260
10.2 Covariance 265
10.3 Correlation 274
10.4 Conditioning on Random Variables 276
10.5 Bivariate Normal Distribution 289
Review Problems 292


► Companion for Chapter 10 299

10B Pattern Appearance 299



► 11 Sums of Independent Random
Variables and Limit Theorems
300


11.1 Moment-Generating Functions 300
11.2 Sums of Independent Random Variables 308
11.3 Markov and Chebyshev Inequalities 314
11.4 Laws of Large Numbers 318
11.5 Central Limit Theorem 321
Review Problems 326



► 12 Stochastic Processes 330

12.2 More on Poisson Processes 330
12.3 Markov Chains 335
12.4 Continuous-Time Markov Chains 353
Review Problems 362


► Companion for Chapter 12 367

12B Brownian Motion 367

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Chapter 1


A xioms of P robability
1.2 SAMPLE SPACE AND EVENTS

1. {M, I, S, P} is a sample space for this experiment, and {I} is the event that the out-
come is a vowel.

2. A sample space is S = {0, 1, 2, . . ., 57}. The desired event is E = {3, 4, 5, 6, 7, 8}.

3. E is the event of at least two heads.

4. E is the event that one die shows three times as many dots as the other. F is the
event that the sum of the outcomes is exactly 6.

5. For 1 ≤ i, j ≤ 3, by (i, j ) we mean that Vann}’s card n u m ber is i, and Paul’s}card
number is j. Clearly, A = (1, 2), (1, 3), (2, 3) and B = (2, 1), (3, 1), (3, 2) .
(a) Since A ∩ B = ∅, the events A and B are mutually exclusive.
(b) None of (1, 1), (2, 2), (3, 3) belongs to A ∪ B. Hence A ∪ B not being the
sample space shows that A and B are not complements of one another.

6. S = {RRR, RRB, RBR, RBB, BRR, BRB, BBR, BBB}.

7. {x : 0 < x < 20}; {1, 2, 3, . . ., 19}.

8. Denote the dictionaries by d 1, d2; the third book by a. The answers are
{d1d2a, d1ad2, d2d1a, d2ad1, ad1d2, ad2d1} and {d1d2a, ad1d2}.

9. EF : One 1 and one even.
EcF : One 1 and one odd.
E c F c : Both even or both belong to {3, 5}.

10. S = {QQ, QN, QP, QD, DN, DP, NP, NN, PP }. (a) {QP };
(b) {DN, DP, NN }; (c) ∅.
}
11. S = x : 7 ≤ x }≤ 961 ; } }
x : 7 ≤ x ≤ 714 ∪ x : 734 ≤ x ≤ 814 ∪ x : 834 ≤ x ≤ 916 .

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12. (a) The sample space is S = {A1A2,1AcA2, A1A2 c, Ac c
A }.
1 2

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

E = {Ac1A2 , A1 A2c , A1c A2c }.

13. S is a sample space for the experiment of flipping a coin until two tails appear con-
secutively.

14. E ∪ F ∪ G = G: If E or F occurs, then G occurs.
EFG = G: If G occurs, then E and F occur.

15. For 1 ≤ i ≤ 3, 1 ≤ j ≤ 3, by aibj we mean passenger a gets off at hotel i
and passenger b gets off at hotel j. The answers are {aibj : 1 ≤ i ≤ 3, 1 ≤ j ≤
3} and
{a1b1, a2b2, a3b3}, respectively.

16. Let a, ℓ, and f represent the outcomes in which the subject identifies almond, lemon,
and flax as his or her favorite color, respectively. Let ∼a, ∼ℓ, and ∼f represent the
outcomes in which the subject does not identify almond, lemon, and flax as his or her
favorite color, respectively. The sample space of the experiment is

S = aℓf, (∼ a)ℓf, a(∼ ℓ)f, aℓ(∼ f ), (∼ a)(∼ ℓ)f, (∼ a)ℓ(∼ f ), a(∼ ℓ)(∼ f ),
}
(∼ a)(∼ ℓ)(∼ f ) .

17. Let x, y, and z be the demand, in thousands, in a random month, for band saws,
reciprocating saws, and hole saws, respectively. The sample space is
}
S = (x, y, z): 30 ≤ x ≤ 36, 28 ≤ y ≤ 33, 300 ≤ z ≤ 600 .

The desired event is
}
(x, y, z) ∈ S : x ≥ 33, z < 435 .

18. Let ad be the outcome that Alexia is dead in five years, and let aℓ be the outcome that
she lives at that time. Define rd and rℓ similarly. A sample space for this experiment
}
is S = (ad, rd), ( ad, rℓ), (aℓ, rd), (}a ℓ, rℓ) . The event that at that time only one
them lives is E = (ad, rℓ), (aℓ, rd) .
of


19. (a) (E ∪ F )(F ∪ G) = (F ∪ E)(F ∪ G) = F ∪ EG.
(b) Using part (a), we have

(E ∪F )(Ec∪F )(E ∪F c ) = (F ∪EEc )(E ∪F c ) = F (E ∪F c ) = FE ∪FF c = FE.

20. (a) ABcCc; (b) A ∪ B ∪ C; (c) AcBcCc; (d) ABCc ∪ ABc C ∪ AcBC;

, Section 1.2 Sample Space and Events 3
(e) ABcCc ∪ AcBcC ∪ AcBCc; (f) (A − B) ∪ (B − A) = (A ∪ B) − AB.




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21. The event that the device is operative at that random time is
n
E= Ai = A1 A2 · · · An.
i=1

22. If B = ∅, the relation is obvious. If the relation is true for every event A, then it
is true for S, the sample space, as well. Thus
S = (B ∩ Sc ) ∪ (Bc ∩ S) = ∅ ∪ Bc = Bc,
showing that B = ∅.

23. Parts (a) and (d) are obviously true; part (c) is true by DeMorgan’s law; part (b) is
false: throw a four-sided die; let F = {1, 2, 3}, G = {2, 3, 4}, E = {1, 4}.

24. Introducing a rectangular coordinate system with origin at the center of dartboard, we
have that a sa m
} ple space for the point at which a dart hits the board is S = (x, y)
2 2
x
: + y < 81 .

}
25. (a) Th e sample space is S = H, TH, TT H
} , T T T H , ... . (b) The desired event is
E = H, TTH, T T T T H, T T T T T T H, ... .
∞ 37
26. (a) n=1
An; (b) n=1
An.
∞
27. Clearly, E1 ⊃ E2 ⊃ E3 ⊃ · · · ⊃ Ei ⊃ · · · . Hence
i=1
Ei = E1 = (−1/2, 1/2).
Now the only point that belongs to all E i’s is 0. For any other point, x, x ∈ (−1, 1),
∞
/ (−1/2i, 1/2i). So
there is an i for which x ∈ i=1
Ei = {0}.

28. Straightforward.

29. Straightforward.

30. Straightforward.

31. Let a1, a2, and a3 be the first, the second, and the third volumes of the dictionary. Let
a4, a5, a6, and a7 be the remaining books. Let A = {a1, a2, . . . , a7}; the
answers are
}
S = x1 x2 x3 x4 x5 x6 x7 : xi ∈ A, 1 ≤ i ≤ 7, and xi /= xj if i /= j
and
}
x1x2x3x4x5x6x7 ∈ S : xixi+1xi+2 = a1a2a3 for some i, 1 ≤ i ≤ 5 ,
respectively.

32. The sample space is
31
S = Ai1Ai2 Ai3 ∪ Aci1Ai2 Ai3 ∪ Ai1Ai2
c
Ai3 ∪ Ai1Ai2Ai3c ∪ A c A c Ai3
i1 i2
c
i=1 A
∪ Aci1 Ai2 A ci3 ∪ Ai1 Aci2 i3 i1 i2 i3

, Section 1.2 Sample Space and Events 5
∪ Ac Ac Ac .




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∞
33. ∞
m=1 n=m
An.

34. Let B1 = A1, B2 = A2 − A1, B3 = A3 − (A1 ∪ A2), . . . , Bn = An −i=1 n−1 Ai,
. . ..



1.4 BASIC THEOREMS

1. No; P (sum 11) = 2/36 while P (sum 12) = 1/36.

2. Since P (AB) = 0, we have 1 ≥ P A ∪ B = P (A) + P (B).

3. 0.33 + 0.07 = 0.40.

4. No, they are not consistent. The first statement implies that the probability of success
is 15/16, while the second statement implies that it is 1/16.

5. Let E be the event that an earthquake will damage the structure next year. Let H
be the event that a hurricane will damage the structure next year. We are given that
P (E) = 0.015, P (H) = 0.025, and P (EH) = 0.0073. Since
P (E ∪ H) = P (E) + P (H) − P (EH) = 0.015 + 0.025 − 0.0073 = 0.0327,
the probability that next year the structure will be damaged by an earthquake and/or a
hurricane is 0.0327. The probability that it is not damaged by any of the two natural
disasters is 0.9673.
Clearly, she made a mistake. Since EF E, we must have P (EF )≤ P (E).
6. ⊆ 1
3 > = P (E).
However, in Tina’s calculations, P (EF )
= 8 4

7. We are inte rested in the pr obabil ity of th e event A ∪ B − AB. Since AB ⊆ A ∪ B,
we have P A ∪ B − AB = P A ∪ B − P (AB) = 0.8 − 0.3 = 0.5.

8. Let A be the event that a randomly selected applicant has a high school GPA of at
least 3.0. Let B be the event that this applicant’s SAT score is 1200 or higher. We
have
P A ∪ B) = P (A) + P (B) − P (AB) = 0.38 + 0.30 − 0.15 = 0.53.
Therefore, 53% of all applicants are admitted to the college.

9. Let J, B, and T be the events that Jacqueline, Bonnie, and Tina win, respectively.
We are given that P (B) = (2/3)P (J) and P (B) = (4/3)P (T ). Therefore, P
(J) = (3/2)P (B) and P (T ) = (3/4)P (B). Now P (J) + P (B) + P (T ) = 1
implies that
3 2
P (B) + P (B)
+