University of Illinois
Fall 2012
ECE 313: Problem Set 13
Law of large numbers, central limit theorem, and joint Gaussian distribution
Due:
Reading:
Reminder:
Wednesday, December 12, at 4 p.m.
ECE 313 Notes Sections 4.10 & 4.11.
The nal exam will be held 1
University of Illinois
Summer 2014
ECE 313: Exam I
Monday, July 7, 2014
7:00 p.m. 8:15 p.m.
269 Everitt Lab
1. (a) Let L denote the restaurant quality level. Then P[L = ] = 1 for = 1, . . . , 5. Let
5
Y denote the Poisson distribution with mean 10 for = 1
University of Illinois
Summer 2014
ECE 313: Exam II
Thursday, July 24, 2014
7:00 p.m. 8:15 p.m.
269 Everitt Lab
1. [10 points] Suppose X has the binomial distribution Bin(n, 1 ) for some n 1.
2
(a) (6 points) Use the Gaussian approximation to nd an approx
University of Illinois
Summer 2014
ECE 313: Exam II
Thursday, July 24, 2014
7:00 p.m. 8:15 p.m.
269 Everitt Lab
1. (a) Note that X Bin(n, 1/2). Let X N (n/2, n/4). It follows from the Gaussian
approximation that
X
0.5 0.1
n

=P

P
P

X 0.5n
n/4
X 0.5
University of Illinois
Spring 2015
ECE 313: Hour Exam I
Wednesday, March 4, 2015
7:00 p.m. 8:15 p.m.
MSEB 100, ECEB 1002, ECEB 2017
1. [20 points] The two parts of this problem are unrelated.
(a) Suppose that you have ve distinct bowls, each can hold up t
University of Illinois
Fall 2011
ECE 313: Midterm Exam I
Monday October 10, 2011
6:40 p.m. 8:00 p.m.
1320 DCL (10 and 11 am sections) and 151 Everitt (noon and 1pm sections)
1. [20 points] Suppose you have three fair dice.
(a) [10 points] If you roll only
University of Illinois
Fall 2011
ECE 313: Hour Exam II
u
1
1. (a) T is an exponential distribution with mean 30 minutes. So fT (u) = 30 e 30 , u 0,
where u is measured in minutes.
(b) Due to the memoryless property of exponential distributions, the waitin
University of Illinois
Fall 2011
ECE 313: Midterm Exam I
Monday October 10, 2011
6:40 p.m. 8:00 p.m.
1320 DCL (10 and 11 am sections) and 151 Everitt (noon and 1pm sections)
1. (a) There are 36 possible outcomes if you roll two dice simultaneously. Of the
University of Illinois
Fall 2011
ECE 313: Hour Exam II
1. [10 points] Buses arrive at a bus station in a Poisson process with an average of 2 buses per
hour.
(a) [5 points] Let T1 be the time of arrival of the rst bus and T2 be the time of arrival
of the
University of Illinois
Summer 2014
ECE 313: Exam I
Monday, July 7, 2014
7:00 p.m. 8:15 p.m.
269 Everitt Lab
1. [12 points] Guessing Restaurant Quality: Suppose you go to a restaurant and do not
know the restaurant quality beforehand. Assume the restaurant
University of Illinois
Fall 2014
ECE 313: Hour Exam I
Wednesday, October 8, 2014
7:00 p.m. 8:15 p.m.
Section B in 151 Everitt Lab, Sections C & E in 141 Loomis Lab, Section D in 100 MSEB
1. [14 points] Suppose n solar cells are in a linear array, in posit
Probability with Engineering Applications
ECE 313 Course Notes
Bruce Hajek
Department of Electrical and Computer Engineering
University of Illinois at UrbanaChampaign
August 2016
c 2016 by Bruce Hajek
All rights reserved. Permission is hereby given to fr
( nk) p (1p)
k
Bernoulli (n , p) p X ( k ) =
nk
pX (k )=
=np
e
k!
k
cfw_
1
,a u b
f
(
u
)
=
Uniform(a , b) x
ba
0 ,else
Exponential( )
E [ X2]
np
Discrete
np(1p)
Discrete
1
p
Discrete
1p
p2
Discrete
Discrete
a+b
2
a +ab+b
3
, for 0 k D
n iscrete
Geometri
University of Illinois
Fall 2014
ECE 313: Hour Exam I
Wednesday, October 8, 2014
7:00 p.m. 8:15 p.m.
Section B in 151 Everitt Lab, Sections C & E in 141 Loomis Lab, Section D in 100 MSEB
1. (a) There are n possibilities for which two cells fail, and n 1 p
University of Illinois
Fall 2014
ECE 313: Hour Exam I
Wednesday, October 8, 2014
7:00 p.m. 8:15 p.m.
Section B in 151 Everitt Lab, Sections C & E in 141 Loomis Lab, Section D in 100 MSEB
1. [14 points] Suppose n solar cells are in a linear array, in posit
University of Illinois
Fall 2014
ECE 313: Hour Exam II
Wednesday, November 12, 2014
7:00 p.m. 8:15 p.m.
Sect. B in 151 Everitt Lab, Sects. C&E in 100 Noyes Lab, Sect. D in 1404 Siebel Center
1. (a) Note that fX (u) = 0.5 for u [0, 1] [2, 3] because the pd
University of Illinois
Fall 2014
ECE 313: Hour Exam II
Wednesday, November 12, 2014
7:00 p.m. 8:15 p.m.
Sect. B in 151 Everitt Lab, Sects. C&E in 100 Noyes Lab, Sect. D in 1404 Siebel Center
1. [24 points] (6 points) Suppose X is a random variable with th
University of Illinois
ECE 313: Final Exam
Fall 2014
Monday, December 15, 2014, 7:00 p.m. 10:00 p.m.
Sect. B, names AO, 1013 ECE, names PZ, 1015 ECE; Section C, names AL, 1015 ECE; all others 112 Gregory
1. (a) On the average, it takes 6 rolls until th
University of Illinois
ECE 313: Conict Final Exam
Fall 2014
Tuesday, December 16, 2014, 7:00 p.m. 10:00 p.m. Room 2013 ECE Building
1. (a) Let c denote the fraction of nonsmoking women who get lung cancer (over some time
period). Then 13c is the fraction
University of Illinois
Spring 2011
ECE 313: Midterm Exam I
Monday February 28, 2011
7:00 p.m. 8:15 p.m.
1NHB 228 in Natural History Building
1. [25 points] Bob ips two fair coins. Let N be the number of heads that are showing. Mary
now draws N cards with
University of Illinois
Fall 2011
ECE 313: Final Exam
1. (a) One way to solve this is using conditional probability using the fact that coin C3 is
grabbed among three coins, while coin C1 is then only grabbed among two coins. Therefore,
P (C3 C1 ) = P (C1
University of Illinois
Spring 2011
ECE 313: Midterm Exam II
Monday April 11, 2011
7:00 p.m. 8:15 p.m.
Sections C and E: 1EVRT 151 in Everitt Lab
Sections D and F: 1GH 100 in Gregory Hall
1. [25 points] Let X be a random variable with CDF:
1
x (, 3]
x
1
University of Illinois
Spring 2009
ECE 313: Hour Exam II
Monday April 13, 2009 7:00 p.m.  8:00 p.m. 100 Noyes Laboratory
1. (a) By DeMorgan's law, Ac B c C c = (ABC)c . Hence, P (Ac B c C c ) = 1  P (ABC), and since A, B, C are mutually independent, we
University of Illinois
Spring 2009
ECE 313: Final Exam
Friday, May 8, 2009, 8:00 a.m. 11:00 a.m. 149 National Soybean Research Center, 1101 W. Peabody Drive, Urbana
1. [20 points] An urn contains ve balls numbered consecutively from 1 to 5. Balls are draw
University
of Illinois
Problem Set #1
Page 1 of 2
ECE 313
Fall 1997
Assigned: Wednesday, September 3, 1997 (or Friday September 5, 1997)
Due:
Wednesday, September 10, 1997
Reading:
Ross, Chapter 1.11.5, Chapter 2.12.5 and 2.7
Noncredit Exercises: (Do not
University
of Illinois
Problem Set #4
Page 1 of 1
ECE 313
Fall 1997
Assigned: Wednesday, September 24, 1997
Due:
Wednesday, October 1, 1997
Reminders: No class on Friday October 3.
Hour Exam I is on Tuesday October 14, 7 pm to 8 pm, in 228
Natural History
University
of Illinois
Problem Set #2
Page 1 of 2
ECE 313
Fall 1997
Assigned: Wednesday, September 10, 1997
Due:
Wednesday, September 17, 1997
Reading:
Ross, Chapter 2.12.5 and 2.7, Chapter 3
Noncredit Exercises: (Do not turn these in) Ross pp. 5461: 3
Axioms and Properties
Events Axioms (F is the set of all events):
E.1 F
E.2 if A FthenAc F
E.3 if A, B, . . . F then A B . . . F
Event Properties:
e.4 F
e.5 if A, B, . . . F then AB . . . F
Probability Axioms:
P.1 For any event A, P (A) 0
P.2 If E1 , E2 ,