University of Illinois
Fall 2014
ECE 313 (Section X)
Homework 10 Solution
Problem 1
a) A valid hazard function h(t) (or z(t) should satisfy the properties in the second column
of the table shown abov
University of Illinois
Fall 2009
ECE 313: Problem Set 4: Solutions Counting Random Variables, Maximum-Likelihood Estimation
1. [Graphical study of binomial pmfs] (a) Work out the numbers yourself. (b)
University of Illinois
Fall 2014
ECE 313 (Section X)
Homework 10
Due Date: Thursday, November 6, 08:00 AM in the class
Write your name and NetID on top of all the pages. Show your work to get partial
ECE 313 - Section X
Midterm Exam Solution
Fall 2014
Name: _
NetID: _
Be sure that your exam booklet has 10 pages.
Write your name at the top of each page.
This is a closed book exam.
You may consult b
Probability with Engineering Applications
ECE 313 Course Notes
Bruce Hajek
Department of Electrical and Computer Engineering
University of Illinois at Urbana-Champaign
August 2014
c 2014 by Bruce Haje
Consider a parallel storage system composed
of nine subsystems, each of which contains
nine servers. Each subsystem can tolerate a
single server failure, and the overall system
can tolerate a single s
University of Illinois
Fall 2012
ECE 313: Problem Set 10
Jointly distributed random variables including independent random
variables
Due:
Reading:
Wednesday, November 7 at 4 p.m.
ECE 313 Course Notes,
University
of Illinois
Problem Set #2: Solutions
Page 1 of 2
ECE 313
Fall 2001
Let f(k) (x) denote the k-th derivative of f(x). Note that f(0) (x) = f(x). Assume that all the derivatives exist.
f(1) (
University of Illinois
Fall 2012
ECE 313: Problem Set 9
Functions of a random variable, failure rate functions, and binary
hypothesis testing for continuous-type observations
Due:
Reading:
Wednesday,
It is much easier to simply add the probabilities in the joint probability matrix than to use
the formula for the probability of error. Since the MAP-decision rule picks the bigger
probability per col
Consider hypotheses H0 and H1 about
a two dimensional observation vector
X = (X1,X2). Under H0, X1 and X2
are mutually independent, and both
have the Poisson distribution with mean
4. Under H1, X1 and
There will be a network outage only if both of the links fail. This is the intersection of link ll
failing and link l2 failing. Suppose A is the event that link [1 fails and B is the event that
link 1
University
Problem Set #4: Solutions
ECE 313
of Illinois
Page 1 of 3
Fall 2001
1.(a) The number of weeks that your investment doubles in value is a binomial random variable Y with
parameters (5,1/2).
University
Problem Set #10: Solutions
ECE 313
of Illinois
Page 1 of 3
Fall 2001
1.(a) The number of arrivals in the interval (0, 4] is the Poisson random variable N(0,4] with parameter 4.
Hence, the m
Hint1
First find the likelihood function of observing (i, j)
under H1 and H0. Then, take their ratio to find
the likelihood ratio A (i, j) = 2E3; , and compare
it with Zl.
Hint2
For (i, j) that the
The entire matrix should add up to 1. 0.2 + 0.3 + 0.1 + 0.1 + 0.1 + 0.2 = 1. That would
mean A + B would equal to 0. It would also mean that A = 0 and B : 0 as well.
Consider the following s tflow network.
The link capacities, in units of some quantity
per unit time, are shown for links that do not
fail. Suppose each link fails with some
probability p and if a lin
University of Illinois
Fall 2012
ECE 313: Problem Set 6
Reliability and cumulative distribution functions (CDFs)
Due:
Wednesday, October 10 at 4 p.m.
Reading:
ECE 313 Course Notes, Sections 2.12 & 3.1
University of Illinois
Fall 2012
ECE 313: Problem Set 4
Geometric distribution, Bernoulli processes, Poisson distribution, ML
parameter estimation, confidence intervals
Due:
Reading:
Wednesday Septemb
University of Illinois
Summer 2017
ECE 313: Exam I
Thursday, July 6, 2017
5.15 - 6.30 p.m.
3015 ECEB
Name: (in BLOCK CAPITALS)
NetID:
Signature:
Instructions
This exam is printed double-sided, so make
University of Illinois
Fall 2017
ECE 313: Hour Exam I
Wednesday, October 11, 2017
8:45 p.m. 10:00 p.m.
Name: (in BLOCK CAPITALS)
NetID:
Section: 2 A, 9:00 a.m.
Signature:
2 B, 10:00 a.m.
2 C, 11:00 a.
University of Illinois Summer 2017
ECE 313: Exam 1
Thursday, July 6, 2017
5.15 6.30 pm.
3015 ECEB
Name: (in BLOCK CAPITALS) 2 C; if g':i \ S j 03
NetID:
Signature:
Instructions
This exam is printed do
University of Illinois Summer 2017
ECE 313: Exam ll
lhursday, July 20, 2017
5.15 6.30 pm.
3015 ECEB
Name: (in BLOCK CAPITALS) S D cfw_v 41 Q; r; S
NetID:
Signature:
Instructions
This exam is close
Bernoulli : distribution of the number of successes on a single Bernoulli trial. We get either a success (p)
or failure (1-p).
Ex:
If a coin is tossed
once, what is the probability it comes up heads?
University of Illinois
Fall 2017
ECE 313: Hour Exam II
Wednesday, November 15, 2017
8:45 p.m. 10:00 p.m.
1. [6 points] Suppose X and Y have the joint pdf:
C u2 + v 2 1
fX,Y (u, v) =
0 else,
(a) Find C
Fall 2017
University of Illinois
ECE 313: Final Exam
Monday, December 18, 2017
1:30 p.m. 4:30 p.m.
1. [14 points] Suppose Y and W are jointly Gaussian random variables with E[Y ] = 2,
E[W ] = 0, Var(Y
University of Illinois
Fall 2017
ECE 313: Final Conflict Exam
Tuesday, December 19, 2017
1:30 p.m. 4:30 p.m.
1. [14 points] Let X be a Gaussian random variable with mean 1 and variance 4. Let Y be a
G
University
of Illinois
Problem Set #5: Solutions
Page 1 of 2
ECE 313
Fall 2001
The maximum-likelihood estimate of p is the observed relative frequency, i.e. p^ = X/N.
Since E[X] = Np and var(X) = Np(1
University
Problem Set #9: Solutions
of Illinois
Page 1 of 2
1.(a) This is a valid pdf.
(b)
This is a valid pdf.
(c)
This is not a valid pdf because the function is negative for 0 < u < 1.
ECE 313
Fal
University
of Illinois
Problem Set #8: Solutions
Page 1 of 4
ECE 313
Fall 2001
With N-ary replication of each component, each replicated component has probability of failure pN since all
the N copies
University
Problem Set #3: Solutions
ECE 313
of Illinois
Page 1 of 2
Fall 2001
1.(a) It is easy to see from the left-hand diagram below that no matter what the outcome is, at least one of A and
B did
University
Problem Set #6: Solutions
ECE 313
of Illinois
Page 1 of 3
Fall 2001
1.(a) X is a binomial random variable with parameters (10, 0.5) and mean 100.5 = 5.
10
10
10
176
848 53
(b)
Pcfw_X 4 = 1