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716
Probability and Random Variables
Appendix B
f (x i )
1
A
3 A A
A
2 2
(a) Uniform PDF
0
A
2
A
3
A
2
x i
f(y 1 ), where y 1 =x 1 +x 2
1
A
3
A
A
A
2 2
0
A
2
A
3
2
A
(b) PDF for y 1 =x 1 +x 2
y 1
3
4A
Exact PDF, f(y 2 ), of
y 2 =x 1 +x 2 +x 3
Gaussian PDF with
1
2ps = 3
4A
3
A
A
A
2 2
0
A
2
A
3
2
A
(c) PDF for y 2 =x 1 +x 2 + x 3 and a Gaussian PDF
y 2
y
Figure B–14
Demonstration of the central limit theorem (Ex. B–8).
PROBLEMS
★ B–1 A long binary message contains 1,428 binary 1s and 2,668 binary 0s. What is the probability of
obtaining a binary 1 in any received bit?
★ B–2 (a) Find the probability of getting an 8 in the toss of two dice.
(b) Find the probability of getting either a 5, 7, or 8 in the toss of two dice.
B–3 Show that
P(A + B + C) = P(A) + P(B) + P(C)
- P(AB) - P(AC) - P(BC) + P(ABC)