, Solutions Manual for
Probability and Random Processes for
Electrical and Computer Engineers
John A. Gubner
University of Wisconsin–Madison
File Generated July 13, 2007
, CHAPTER 1
Problem Solutions
1. Ω = {1, 2, 3, 4, 5, 6}.
2. Ω = {0, 1, 2, . . . , 24, 25}.
3. Ω = [0, ∞). RTT > 10 ms is given by the event (10, ∞).
4. (a) Ω = {(x, y) ∈ IR2 : x2 + y2 ≤ 100}.
(b) {(x, y) ∈ IR2 : 4 ≤ x2 + y2 ≤ 25}.
5. (a) [2, 3] c = (−∞, 2) ∪ (3, ∞).
(b) (1, 3) ∪ (2, 4) = (1, 4).
(c) (1, 3) ∩ [2, 4) = [2, 3).
(d) (3, 6] \ (5, 7) = (3, 5].
6. Sketches:
y y y
1
1 −1
x x x
−1
B0 B1 B−1
y y y
3
x x x
3
C1 H3 J3
1
, 2 Chapter 1 Problem Solutions
y y
3
3
x x
3 3
U
H3 J3 = M 3 H3 U J3 = N 3
y
y 4
2 3
x x
2 3 4
U U
M2 N 3 = M2 M4 N3
7. (a) [1, 4] ∩ [0, 2] ∪ [3, 5] = [1, 4] ∩ [0, 2] ∪ [1, 4] ∩ [3, 5] = [1, 2] ∪ [3, 4].
(b)
c
[0, 1] ∪ [2, 3] = [0, 1] c ∩ [2, 3] c
h i h i
= (−∞, 0) ∪ (1, ∞) ∩ (−∞, 2) ∪ (3, ∞)
h i
= (−∞, 0) ∩ (−∞, 2) ∪ (3, ∞)
h i
∪ (1, ∞) ∩ (−∞, 2) ∪ (3, ∞)
= (−∞, 0) ∪ (1, 2) ∪ (3, ∞).
∞
\
(c) (− 1n , n1 ) = {0}.
n=1
\∞
1
(d) [0, 3 + 2n ) = [0, 3].
n=1
[∞
1
(e) [5, 7 − 3n ] = [5, 7).
n=1
[∞
(f) [0, n] = [0, ∞).
n=1