Section 2: Probability Theory and Random Processes
ECE 316

Spring 2008
ECE316: Tutorial 1
1
Problems
1. How many numbers lying between 100 and 1000 can be formed using the digits 0, 1, 2, 3, 4, 5, if repetition
of digits is not allowed?
2. Find the number of arrangements
Section 2: Probability Theory and Random Processes
ECE 316

Spring 2008
ECE316: Tutorial 3 (May 24, 2017)
1
Problems
1. Prove that P (
i=1 Ai )
P
i=1
P (Ai ).
2. Show that if P (Ai ) = 1 for all i 1, then P (
i=1 Ai ) = 1.
3. If 5 people are present in a room, what is th
Section 2: Probability Theory and Random Processes
ECE 316

Spring 2008
ECE316: Tutorial 5 (June 7, 2017)
1
Problems
1. Show that, for any events E and F, P (EE F ) P (EF ).
2. Show that if P (AB) = 1, then P (B c Ac ) = 1.
3. There is a 60 percent chance that event A
Section 2: Probability Theory and Random Processes
ECE 316

Spring 2008
ECE316: Tutorial 4 (May 31, 2017)
1
Problems
1. Prove that if P (A) > 0, then P (ABA) P (ABA B).
2. Mother, father and son line up at random for a family picture. Let E: son on one end and F : fathe
Section 2: Probability Theory and Random Processes
ECE 316

Spring 2008
ECE316: Tutorial 2 (May 17, 2017)
1
Mutinomial Theorem
1. In the 1st round of a knockout tournament involving n = 2m players, the n players are divided in n2
pairs. Each pair plays a game. The losers
Section 2: Probability Theory and Random Processes
ECE 316

Spring 2008
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Section 2: Probability Theory and Random Processes
ECE 316

Spring 2008
UNIVERSITY OF WATERLOO
DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
ECE 316 Probability Theory and Random Processes
Midterm Examination
Wednesday February 13, 2008, 5:30 pm  7:00 pm
Instructor:
Section 2: Probability Theory and Random Processes
ECE 316

Spring 2008
Problems
3. FIX and Y two independent uniform {0,1} random ohrs'ohtes, show that
2
{o+1)[os+2} for a} 0'
Er
X _ ylral =
Solution. From the denition of the expectation, we hove
I. I.
XY*]=L A Ixelde
Section 2: Probability Theory and Random Processes
ECE 316

Spring 2008
Problems
1. Two fair dice we miied. Find the joint pivbubiiy mass meiion of X and V when
(a) X is the Imyesi value obtained an any die and Y is the sum a] the wines.
Solution. (3.) The joint mass func
Section 2: Probability Theory and Random Processes
ECE 316

Spring 2008
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Section 2: Probability Theory and Random Processes
ECE 316

Spring 2008
Problems
1. (a) How manyI dierent 7plaoe license plates are possible if the rst 2 places are for letters
and other 5 for numbers?
Solution. By the generalized version of the basic principle the answer
ECE316 NotesWinter 2017: A. K. Khandani
1
1.1
1
Combinatorial Analysis
Introduction
This chapter deals with finding eective methods for counting the number of ways that things can occur.
In fact, man