Random Variables and Stochastic Processes

Homework 1 Solutions
7CCSMRVA - Random Variables and Stochastic Processes
October 5, 2018


Problem 1. If A = {2  x  5} and B = {3  x  6}, find A [ B, AB and (A [ B)(AB).
Please note that (AB)c can also be denoted as (AB).

Solution 1.

A [ B = {2  x  5} [ {3  x  6} = {2  x  6} (1)
AB = {2  x  5} \ {3  x  6} = {3  x  5} (2)
AB = {{x < 3} [ {x > 5}} (from (2)) (3)
(AU B)(AB) = {2  x  6} \ {{x < 3} [ {x > 5}} (4)
= {2  x < 3} [ {5 < x  6} (5)

Problem 2. We select at random m objects from a set S of n objects and we denote by
Am the set of the selected objects. Show that the probability that a particular element ⇣0 of
S is in Am equals to p = m/n.

Solution 2. There are two solutions to this problem.
First solution:
Using the concept of combinations, we have

# of m-element subsets containing ⇣0
p= (6)
# of m-element subsets
✓ ◆
n 1
(n 1)!
m 1 (n m+1)!(m 1)! m
= ✓ ◆ = n!
= (7)
n m!(n m)!
n
m

Second solution:
m
Probability that ⇣0 belongs to a specific Am is clearly P (⇣0 |Am ) = n
. Then, we sum all
over sets Am

X X m m X m m
p= P (⇣0 |Am )P (Am ) = P (Am ) = P (Am ) = ·1= (8)
all sets Am all sets Am
n n all sets A n n
m




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