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 den
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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
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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 qui
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 ha
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
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.
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
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
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
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). Fi
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
Collect
Cumulative Distribution Function
CDFs are defined for any RV (continuous or discrete)
For continuous case
For discrete case
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CDF Properties
It is the probability of something, so between 0 and 1
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:
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Countable Infinity
4
Countable Infinity
EEE 304: WEEK 1 PRACTICE PROBLEM SOLUTIONS
Problem 1:
Consider the filter with impulse response h(t )
= e 2t u (t 2) .
1. Find the transfer function
2. Find the Laplace transform of the output when x
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
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 ma
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 someth
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
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Definition
PMF of Number of Arrivals
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 (w
Experiment
A repeatable procedure that produces
potentially different outcomes at every trial
Set of all outcomes is the sample space
Events are sets of outcomes
Events are subsets of the sample s
Random Variables
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Random Variables Terminology
RVs are just tagging numerical values to experiments.
The probability assignment to outcomes can be expressed
in terms of the RVs.
There are differen
Continuous Case
1
Example
2
Example: Squaring a Uniform RV
X is a uniform RV over (-1,1), and
Note that W is always positive, even though X isnt
3
Example: Square Root
RV
; X is exponentially distr
Joint Distributions: CDF
Applies to continuous or discrete
Can be extended to more than 2 RVs
CDF factors to product of marginals if RVs independent
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Example
2
Solution
Both X and Y are uniformly
Summarizing and Visualizing Data
Problem: Big data and often no given (probability) model
Convenient to summarize the data by processing it
Most common examples are
Sample mean (average)
Sample varian
Variance
Measure of spread of the PMF
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Variance Examples
Consider the effect of parameters on variance
Random Variable
Variance
Bernoulli
Geometric
Binomial
Pascal
Poisson
Discrete Uniform
Proofs re
Covariance
Can be computed if joint PDF is given
Measures if RVs are related in some way
Also comes out naturally from variance of a sum
Can be normalized with std (corr. coeff)
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Covariance
2
Covaria
Introduction to Classical Statistics
Unlike Bayesian case,
is not a conditional PDF.
Unlike Bayesian case, there is no prior on
contains all unknown (deterministic) parameters
contains all N measureme
Expectation
AKA mean of the RV or the distribution
Interpretations
Center of gravity of the PMF
Average in large number of repetitions of the experiment
1
Example: Uniform PMF
2
Example: Geometric
EEE 350 Random Signal Analysis
Homework Assignment #4 Solution
Assigned: 22 October 2017
Due: 31 October 2017
1. X and Y are independent with
PX (x) =
75
x
75
(1/2)
and PY (y) =
25
y
(1/2)25
2. (a
EEE 350 Random Signal Analysis
Homework Assignment #5
Assigned: 04 November 2017
Due: 14 November 2017
Reading (from Yates & Goodman):
Chapter 6
Instructions:
1. Homework must be submitted on paper, e
EEE 350 Random Signal Analysis
Homework Assignment #3
Assigned: 3 October 2017
Due: 12 October 2017
Reading (from Yates & Goodman):
Chapter 5
Instructions:
1. Homework must be submitted on paper, eith
EEE 350 Random Signal Analysis
Homework Assignment #3 Solutions
Assigned: 3 October 2017
Due: 12 October 2017
1. Each test of an integrated circuit produces an acceptable circuit with probability p, i
EEE 350 Random Signal Analysis
Homework Assignment #4
Assigned: 22 October 2017
Due: 31 October 2017
Reading (from Yates & Goodman):
Chapter 5
Instructions:
1. Homework must be submitted on paper, eit