EE 562a Random Processes in Engineering

EE 562a: Random Processes in Engineering
EE department, USC, Fall 2014
Instructor: Prof. Salman Avestimehr

Homework 1
Solutions

1. (Axioms of Probability) Prove the union bound: n P [∪n Ak ] ≤ k=1 P [Aj ]. j=1 The union bound is useful because it does not require that the events Aj be independent or disjoint. Problem 1 Solution
We prove this part by induction, for k = 2 we have
P (A1 ∪ A2 ) = P (A1 ) + P (A2 ) − P (A1 ∩ A2 ) ≤ P (A1 ) + P (A2 )

(1)

Now, assume that the statement is true for k = n n P (A1 ∪ A2 . . . ∪ An ) ≤

P (Ai )

(2)

i=1

For k = n + 1, we can write
P (A1 ∪ A2 . . . ∪ An+1 ) = P (An+1 ) + P (A1 ∪ A2 . . . ∪ An ) − P (An+1 ∩ (A1 ∪ A2 . . . ∪ An )) n+1 ≤

P (Ai ) i=1 (3)
This proves the union bound.
2. (Independence) For each one of the following statements, indicate whether it is true or false, and provide a brief explanation.
(a) If P (A|B) = P (A), then P (B|Ac ) = P (B).
(b) If 5 out 10 independent fair coin tosses resulted in tails, the events “first toss was tails” and “10th toss was tails” are independent.

(c) If 10 out 10 independent fair coin tosses resulted in tails, the events “first toss was tails” and “10th toss was tails” are independent.
Problem 2 Solution
(a) True.
P (A|B) = P (A), therefore,
P (A ∩ B) = P (A)P (B)

(4)

But we know that (A∩B) and (Ac ∩B) are disjoint events, and (A∩B)∪(Ac ∩B) = B.
Therefore,
P (A ∩ B) + P (Ac ∩ B) = P (B)
(5)
Thus, from (4) and (5) we can write
=⇒
=⇒

P (B) − P (Ac ∩ B) = P (A)P (B)
P (Ac ∩ B) = (1 − P (A))P (B) = P (Ac )P (B)
P (Ac )P (B)
= P (B)
P (B|Ac ) =
P (Ac )

(b) False.
Let A designate the event “5 out of 10 independent fair coin tosses are tails”.
Also, let B designate the event “first toss of the fair coin results in tails”.
In addition, let C designate the event “10th toss of the fair coin results in tails”.
We have
P (B|A) =

1
2

P (C|A) =
But,

P (A ∩ B)
=
P (A)
P (A ∩ C)
=
P (A)