The Fourier Domain
A complex study of complexity
Fourier Domain
Expresses an image as the sum of weighted sinusoids
Wavelengths are determined by image dimensions
Amplitudes are determined by sample values
Fourier coefficients are complex rather than real
EE 178
Probabilistic Systems Analysis
Autumn 2016 Tse
Lecture Note 13
Continuous Probability Continued
In Lecture 11, we introduced continuous random variables. For example, lets consider X Uni f [0, 1]
which is a random variable that takes on continuous
EE 178
Probabilistic Systems Analysis
Autumn 2016 Tse
Lecture Note 12
Continuous random variables
Up to now we have focused exclusively on discrete random variables, which take on only a finite (or countably infinite) number of values.
But in real life ma
EE178: Probabilistic Systems Analysis, Autumn 2016
Homework 6
Due Wednesday , Nov 9, 5pm
1. Estimating Variance
You have available n samples X1 , X2 , . . . , Xn , drawn independently from a pmf pX . Suppose you know the
mean but not the variance 2 of the
HW1 Solutions
October 5, 2016
1.
(20 pts.) Random variables, sample space and events
Consider the random experiment of ipping a coin
1.
3.
ith
coin ip
HHHH
HTHH
THHH
TTHH
HHHT
HTHT
THHT
TTHT
HHTH
HTTH
THTH
TTTH
HHTT
HTTT
THTT
TTTT
(2 pts.)
A.
i cfw_1, 2,
EE178 HW 8 Solutions
Nov 30, 2016
1.
(15 pts) Gaussian Distribution
If a set of grades on a probability examination in an inferior school (not Stanford!) are approximately Gaussian distributed with a mean of 64 and a standard deviation of 7.1, nd:
5% of t
EE 178
Probabilistic Systems Analysis
Autumn 2016 Tse
Lecture Note 4
Conditional Probability Review
In the previous lecture, the conditional probability P(A|B) was defined as
summarizes the differences before and after conditioning on the event B.
P(AB)
P
EE178: Probabilistic Systems Analysis, Autumn 2016
Homework 1
Due Wed, Oct 5, in class
1. Random variables, sample space and events
Consider the random experiment of flipping a coin 4 times.
(a) Define the appropriate random variables.
(b) List all the ou
EE 178
Probabilistic Systems Analysis
Autumn 2016 Tse
Lecture Note 15
Gaussian Distribution
The last continuous distribution we will look at, and by far the most prevalent in applications, is called the
Normal or Gaussian distribution. It has two paramete
HW4 Solutions
1. (20 pts.) Packets Over the Internet Again
n packets are sent over the Internet (n even). Consider the following probability models for the process:
(a) Each packet is routed over a different path and is lost independently with probability
EE 178
Probabilistic Systems Analysis
Autumn 2016 Tse
Lecture Note 18
Recap
In the general model of prediction, we have a model (with parameter ). We collect data to estimate the
parameters of the model. After estimating the parameters, we can use the mod
HW5 Solutions
1. (50 pts.) Random homeworks again
(a) (8 pts.) Show that if two random variables X and Y are independent, then
E[XY ] = E[X]E[Y ]
Answer: Applying the definition of expectation we have
XX
E[XY ] =
xy pX,Y (x, y)
x
=
y
XX
x
xy pX (x)pY (y)
Plotting a Confidence Band Over a Scatterplot With Regression Line
Assume you have the data set named Data from Problem 1.19, with explanatory variable
named ACT and response variable named GPA. Assume further that you have fit a linear
model to the data,
Addendum for Spring Quarter, 2014
Newly Added:
MEDICINE (MED) 120N | 3 UNITS |
Pathophysiology of Disease of the Heart
Preference to Freshmen. This course presents the anatomy and physiology of the heart in normal and in disease states. It clarifies the
u
An Introduction
to
Social Psychology
William McDougall, D.Sc., F.R.S.
Fellow of Corpus Christi College, and Reader in
Mental Philosophy in the University of Oxford
Fourteenth Edition with Three Supplementary Chapters
Batoche Books
Kitchener
2001
William M
Confidence Intervals
Objectives:
Students should know how to calculate a standard error,
given a sample mean, standard deviation, and sample
size
Students should know what a confidence interval is, and
its purpose
Students should be able to construct and
Hardegree, Modal Logic; c6: Modal Predicate Logic
6
A.
B.
C.
D.
27
VI-1
Modal
Predicate Logic
Ordinary Predicate Logic .2
1.
Introduction.2
2.
Noun Phrases.2
3.
Predicates .3
4.
Quantifiers as Noun Phrases.5
5.
A Problem with the Subject-Predicate Analysi
Lecture 6. Entropy of an Ideal Gas (Ch. 3)
Today we will achieve an important goal: well derive the equation(s) of state
for an ideal gas from the principles of statistical mechanics. We will follow the
path outlined in the previous lecture:
Find (U,V,N,.
A note on how well be evaluating your papers. Admittedly, grades on philosophical
essays are less objective than grades for physics problem sets, say (though, we
like to think, more objective than, for example, grades for creative writing
assignments or,
How many ways are there to pass through city A where the
arrows represent one-way streets?
Answer: mn ways
The counting principal: Suppose two experiments are to
be performed. If experiment 1 can result in any of m
m roads
A
.
n roads
.
possible outcome
Bayes' Rule
Bayes' Rule - Updating Probabilities
Let A1,.,Ak be a set of events that partition a sample space such that (mutually exclusive and exhaustive):
each set has known P(Ai) > 0 (each event can occur) for any 2 sets Ai and Aj, P(Ai and Aj) = 0 (
EE 178
Probabilistic Systems Analysis
Autumn 2016 Tse
Lecture 9
Some Important Distributions
There are four important distributions in probability: binomial, geometric, Poisson and Gaussian. We have
covered the first one. We will now cover the second one.
EE 178
Probabilistic Systems Analysis
Autumn 2016 Tse
Lecture Note 1
Introduction to Probability1
Life is full of uncertainty.
Probability is a framework to deal with uncertainty. Probability theory has its origins in gambling analyzing card games, dice,
HW3 Solutions
1. (20 pts.) Packets Over the Internet
n packets are sent over the Internet (n even). Let Xi = 1 if the ith packet got lost and Xi = 0 otherwise.
Consider the following probability models for the packet loss process:
(i) Each packet is route
HW2 Solutions
1. (13 pts.) Colorful coins
(a) (3 pts.) Describe the basic random variables and the outcomes in the sample space, and give their
probabilities.
Answer: We have one random variable C which denotes the coin chosen (1, 2 and 3, with 1 being th
EE 178
Probabilistic Systems Analysis
Autumn 2016 Tse
Lecture 5
Independent random variables and building probability models
Just a recap, there are 3 steps in building a probability model of a real-world problem:
1. Identify the basic random variables.
2
EE178: Probabilistic Systems Analysis, Autumn 2016
Homework 4
Due Wednesday Oct 26, 5pm (Note new time.)
1. Packets Over the Internet Again
n packets are sent over the Internet (n even). Consider the following probability models for the packet loss
proces
EE 178
Probabilistic Systems Analysis
Autumn 2016 Tse
Lecture Note 7
Examples of distributions continued
2 The homeworks of n students are collected in, randomly shuffled and returned to the students. What
is the distribution of the number of students who
EE178: Probabilistic Systems Analysis, Autumn 2016
Homework 2 (Alternate)
Due Wed. Oct. 12 in class
1. Colorful coins
We are given three coins. The first coin is a fair coin painted blue on the heads side and white on the tails
side. The other two coins a