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Probability Homework Solutions

1) This homework assignment involves solving probability and random process problems from three textbooks and proving properties about one-to-one correspondences between sets. 2) It also involves showing that the sample space of an infinite coin toss sequence and the set of all subsets of natural numbers are uncountably infinite, and analyzing countable and uncountable properties of rational numbers, irrational numbers, and real numbers. 3) Finally, it asks to prove that a countable union of countable sets is countable, and to provide an example of a field that is not a σ-field by considering subsets of the natural numbers.

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Rahul Agrawal
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51 views2 pages

Probability Homework Solutions

1) This homework assignment involves solving probability and random process problems from three textbooks and proving properties about one-to-one correspondences between sets. 2) It also involves showing that the sample space of an infinite coin toss sequence and the set of all subsets of natural numbers are uncountably infinite, and analyzing countable and uncountable properties of rational numbers, irrational numbers, and real numbers. 3) Finally, it asks to prove that a countable union of countable sets is countable, and to provide an example of a field that is not a σ-field by considering subsets of the natural numbers.

Uploaded by

Rahul Agrawal
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Homework 2
Probability and Random Processes (EE 325), Spring’18

1) The following problems from Papoulis: 2.1, 2.2, 2.3, 2.4, 2.5, 2.7 (on p. 44).
2) The following problems from Stark and Woods: 1.4 (on p. 48), 1.8 (on p. 49).
3) The following problems from Grimmett and Stirzaker: 1.2.4 (on p. 1), 1.8.3 (on p. 4).
4) If a one-to-one correspondence exists from set A to set B, then we write A ∼ B. Show
that:
a) A ∼ A.
b) If A ∼ B, then B ∼ A.
c) If A ∼ B and B ∼ C, then A ∼ C.
(That is, the one-to-one correspondence relation is reflexive, symmetric and transitive.)
5) (a) A coin is tossed an infinite number of times. The sample space Ω is defined to be the
set of all sequences a1 a2 a3 . . ., where aj ∈ {H, T } for j = 1, 2, 3, . . .. Show that Ω is
an uncountable set.
(b) Show that the set of all subsets of the set of natural numbers is an uncountable set.
(Hint: For both parts, try the technique we used in class to show that the set [0, 1] is
uncountable.)
6) (a) Show that the set of all rational numbers is countably infinite.
(b) Show that the set of all irrational numbers is uncountably infinite.
(c) Let R denote the set of real numbers. Let A be any subset of R that is a superset of
some interval [a, b], where a < b. Show that A is an uncountable set. Deduce that each
interval of the form [a, b) or (a, b] or (a, b), where a < b, is an uncountable set.
(d) Show that for each integer n ≥ 1, the set Rn is an uncountable set.
7) Let E1 , E2 , E3 , . . . be a sequence of countably infinite sets and let S = ∪∞
n=1 En . Prove that

S is countably infinite.
8) a) Let Ω be any finite set. Show that every field of subsets of Ω is a σ-field.
2

b) Let Ω = {1, 2, 3, . . .}. Let:

A1 = {{n, n + 1, n + 2, . . .} : n = 1, 2, 3, . . .},

A2 = {A : A ⊆ Ω, A finite},

and
A3 = {A1 ∪ A2 : A1 ∈ A1 , A2 ∈ A2 }.

Let
F = A1 ∪ A2 ∪ A3 .

Show that F is a field, but not a σ-field. (Hint: For showing that F is not a σ-field,
consider the set {1, 3, 5, . . .}.)

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