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 of letters of the word INDEPENDENCE. In how many arran
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 the probability that
(a) no two of them have the same bir
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 will occur. If A doesnt occur, then there is a 10 perc
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 : father
in middle. Find P (EF ).
3. Six balls are to be rand
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 of the games are eliminated while the winners go on to
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: L.L. Xie
Time of exam: 5:30 pm
Duration of exam: 90 mi
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 Ixeldeds.
Next, we compute the inner integration as foUcuws:
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 function of X and V1 p[i1j} = P[X = LY = can be
omnputenl
Section 2: Probability Theory and Random Processes
ECE 316

Spring 2008
Mm 170 mm
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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 is
26261010101010=67,500,000
(b) Repeat part (a
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, many problems in probability theory can be solved simply b