1
EEE 350 Fall 2015
HW 2
Due at 3pm on Sept. 21
(1) Roll a 4-sided die twice and assume all sixteen outcomes are equally likely. Consider the following
events:
A : The difference of the two numbers is 2 (Let x denote the first number and y denote the
seco
Homework 3 Solutions
Problem Solutions : Yates and Goodman, and 2.6.4 2.2.5 2.3.4 2.3.6 2.4.2 2.4.5 2.5.6 2.5.7 2.6.3
Problem 2.2.5 Solution
Using B (for Bad) to denote a miss and G (for Good) to denote a successful free throw, the sample tree for t
1
Practice Quiz 2 EEE 350, Spring 2014
Answer all 5 questions.
Time for test = 60 minutes.
Use of calculator is permitted. No notes allowed.
Gaussian CDF tables will be provided for the in-class quiz.
1) If the density function of X is
ce2|x|
0
fX (x) =
Midterm BEE 350, March 5th 2014
Name in capitals as it appears on the roster:
. Answer all questions
. Time for test = 75 minutes.
0 Use of calculator is permitted.
- A single A4 sheet (both sides) of notes is permitted.
1) Consider two independent coin t
Homework 5 Solutions
Problem Solutions : Yates and Goodman, 3.4.2 3.4.5 3.5.3 3.5.5 3.6.1 and 3.6.4
Problem 3.4.2 Solution
From Appendix A, we observe that an exponential PDF Y with parameter > 0 has PDF fY (y) = e-y y 0 0 otherwise (1)
In additi
Probability and Stochastic Processes:
A Friendly Introduction for Electrical and Computer Engineers
Edition 2 Roy D. Yates and David J. Goodman
Problem Solutions : Yates and Goodman, 3.5.3 3.5.4 3.5.7 3.5.8 3.5.10 3.7.2 3.7.5 3.7.7 3.7.11 and 3.7.16
EEE 350 Random Signal Analysis
Midterm Exam II
November 16, 2015
Welcome to the second midterm examination! Please read everything on this page before you begin.
1. As you should already know, you may not consult any materials during the exam other than t
Probability and Stochastic Processes:
A Friendly Introduction for Electrical and Computer Engineers
Edition 2 Roy D. Yates and David J. Goodman
Problem Solutions : Yates and Goodman, 1.9.4 1.10.1 and 1.10.2 1.5.5 1.6.4 1.7.5 1.8.1 1.8.3 1.8.4 1.8.6
Probability and Stochastic Processes:
A Friendly Introduction for Electrical and Computer Engineers
Edition 2 Roy D. Yates and David J. Goodman
Problem Solutions : Yates and Goodman, 1.5.2 and 1.5.3 1.2.1 1.2.2 1.2.4 1.3.1 1.3.2 1.3.4 1.4.1 1.4.4
P
Homework 8 Solutions
Problem Solutions : Yates and Goodman, 4.8.3 4.8.4 4.8.6 4.9.3 4.9.4 4.10.5 and 4.10.8
Problem 4.8.3 Solution
Given the event A = {X + Y 1}, we wish to find f X,Y |A (x, y). First we find
1 1-x 0
P [A] =
0
6e-(2x+3y) dy dx =
Counting!
1
Discrete Uniform Law
2
Basics
3
Example
0.0154
Dont bet your money on it!
4
Combinations
5
Sum of Combinations
Proof: Counting bit strings of length n
kth term of the sum = # of length n-bit strings with k ones
Both sides equal to total #
Example
Experiment: single dice roll
Outcomes are just 1,2,3,4,5,6.
Events are the set of all subsets
Closed under unions and intersections
There are 64 events
The probability of an event = (# of elements in the event) x 1/6
E.g.,
1
Twist on Example
Same
EEE 350: Random Signal
Analysis
1
Example: Birthday Problem
Birthday problem: Given N people in a room, what is
the probability that 2 or more people will have the
same birthday?
1 Probability that all N people have different bdays
2
Example: Birthday Pro
EEE304
Lecture 1.2: Review of basic signals and
their properties: Exponentials
Exponentials
CTExponential: e st e t jt e t e jt e t (cos t j sin t )
When =0,theexponentialisperiodicwithperiod2/
When =0,theexponentialhasmagnitude1:
e jt cos 2 t sin 2 t 1
D
EEE304
Lecture 1.1a: Review of basic signals
and their properties: Steps and Impulses
Unit Step
if t 0
1
0 otherwise
Unitstep: u (t )
Theunitstepservesasasetindicator,i.e.,whetheranargumentbelongstoasetornot.
Itisusefulinwritingcompactexpressionsforruleb
EEE304
Week 1: Review of Signals and Systems
Fundamental Concepts
X
H
Y
EEE304
Lecture 1.1a: Review of basic signals
and their properties: Steps and Impulses
Unit Step
Unit step:
if t 0
1
u (t ) =
0 otherwise
The unit step serves as a set indicator, i.e.
EEE304
Lecture 1.1b: Review of basic systems
and their properties: Impulse response
Impulses and LTI systems (CT)
FortheparametrizationoftheoutputofanLTIsystemintermsofshifted
impulseresponses(convolutionintegral)weoperatewiththesystemonx:
H [ x(t )] H
Random Vectors (Multivariate)
Discrete:
Continuous:
1
I.I.D. Multivariate Random Vectors
2
Example
3
Distribution of max
CDF of the max of n i.i.d. RVs is the nth power of the CDF of one:
Example: What is the PDF of the max of n i.i.d. exponential RVs?
PD
Review of Set Theory
Probability theory is grounded in set theory
Union
Intersection
Compliment
De Morgans Law
1
Set Union
2
Set Intersection
3
Set Compliment
4
Mutually Exclusive (Disjoint)
5
Collectively Exhaustive
6
Other Basic Definitions
means if
the
Cumulative Distribution Function
CDFs are defined for any RV (continuous or discrete)
For continuous case
For discrete case
1
CDF Properties
It is the probability of something, so between 0 and 1
Monotone non-decreasing function
Defined both for con
Geometric Sum Formula
Will be useful throughout the class
What happens when |r|1 ?
What happens when we differentiate both sides wrt r ?
Alternative way:
1
Countable Infinity
4
Countable Infinity Cont
Can verify
using the geometric sum formula (r=1/2
Law of Large Numbers and the
Central Limit Theorem
The LLN and CLT are about convergence of sums of a large # of RVs
LLN is about the average (sum / # of RVs)
CLT involves a different normalization (with the sqrt of the # of RVs)
Both are fundamental in p
Independence of Events
1
Independence is Different
than Disjointness
Disjoint:
If
disjoint,
If disjoint, events cannot occur at same time.
Independence:
If independent, occurrence of one event has no
influence on the occurrence of the other.
Independ
Bernoulli PMF
Bernoulli(p) RV takes on values 0 or 1
Same RV regardless of experiments producing
cfw_0,1
cfw_H,T
cfw_Accept,Reject
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Binomial PMF
n indep. coin flips, p is prob. of success; how many successes?
Binomial (n,p) RV takes on values cfw_0,1
Probabilities with cards
Discrete uniform law
Example: Probability of drawing an ace
Splitting Deck into 4; one Ace Each
= 10.5%
2
Four of a kind
A hand in poker
Four cards of same rank and something
Full House
Three cards of same rank + two other ca
Poisson Process
Continuous analogue of Bernoulli process
Can be described as points on a line
OR, as an increasing staircase that goes up
by one at every point
1
Definition
PMF of Number of Arrivals
Number of arrivals during an interval of length t is
3
Lecture 1.1a: Review of basic signals and their properties: Steps and Impulses
Unit step
The unit step is one of the most frequently used functions in the study of LTI systems
and systems in general. It is important because it is an easy-to-implement test
EEE304
Lecture 1.5a: Properties of CT LTI
systems in the Frequency Domain: ROC
System properties from the transfer function:
Region of Convergence
ThedevelopmentofequivalentconditionsforthepropertiesofLTIsystems
fromitstransferfunction(TF)hingesontheconce