EE 505 B, Autumn, 2011
Two or More Random Variables
1
1 Two Random Variables
1.1 Finding the joint cdf from the joint pdf
Example 1.1. Find the cdf of the pdf we used in class. The pdf is
f X1 X2 ( x 1 , x 2 ) =
2
0 x2 x1 1
0
all other values of x1 and x2
List of trigonometric identities - Wikipedia, the free encyclopedia
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List of trigonometric identities
From Wikipedia, the free encyclopedia
In mathematics, trigonometric identities are
equalities that involve trigonometric functions and
are tr
List of integrals of trigonometric functions - Wikipedia, the free encyclopedia
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List of integrals of trigonometric functions
From Wikipedia, the free encyclopedia
The following is a list of integrals (antiderivative
functions) of trigonometri
The Eigenvalue Problems - 8.8
1. Definition of Eigenvalues and Eigenvectors:
Let A be an n ! n matrix. A scalar ! is said to be an eigenvalue of A if the linear system Av " !v has a
nonzero solution vector v. The solution vector v is said to be an eigenve
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EE 505 B Fall 2011
Final Exam Solutions
1. [35 points]
(a) [10 points] We use the formula
f X (x) =
f U (u)
d
du g(u) u = u i
i
x
There is only one solution u1 that sat1
ises the equation x = ln(u):
x
u1 = e x
ex
Substituting u1 into the pdf:
f U ( u1 ) =
EE 505 B Fall 2011
Final Exam
Nota bene: For full credit you must show all of the steps or reasoning that you used to
get the answer.
1. [35 points] U is a uniformly distributed random variable with probability density
function
1 0u1
f U (u) =
0 for all o
453.701 Linear Systems, S.M. Tan, The University of Auckland
9-1
Chapter 9 The Discrete Fourier transform
9.1
Denition
When computing spectra on a computer it is not possible to carry out the integrals involved in
the continuous time Fourier transform. In
Euler's Identity
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Euler's Identity
Euler's identity (or `theorem' or `formula') is
(Euler's Identity)
To `prove' this, we will first define what we mean by `
, is assumed
EIGENVALUES AND EIGENVECTORS
1. Definition
They are dened in terms of each other. Let A be an n n matrix. A vector
v = 0 is an eigenvector of A with eigenvalue if the equation
Av = v
is satised.
Note that eigenvectors are not uniquely dened:
If v is an ei
EE 500 PMP Colloquium/Seminar - UWEE - EE 500
Academics
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Autumn 2011
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Section 17.9 The Divergence Theorem
Turning a Flux Integral into a Triple Integral The last result we consider is a generalization of Greens Theorem converting a ux integral over a closed surface into a triple integral over the interior of the surface. 1.
Divergence Theorem Examples
Gauss' divergence theorem relates triple integrals and surface integrals.
GAUSS' DIVERGENCE THEOREM Let F be a vector field. Let W be a closed surface, and let e be the region inside of W . Then: ( ( F . A oe ( ( ( divaFb .Z
W
DFT Definition
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DFT Definition
The Discrete Fourier Transform (DFT) of a signal
where `
may be defined by
' means `is defined as' or `equals by definition', and
The sampl
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1.4 The Matrix Equation Ax b
Linear combinations can be viewed as a matrix-vector
multiplication.
Definition
If A is an m n matrix, with columns a 1 , a 2 , , a n , and if x is in
R n , then the product of A and x, denoted by Ax, is the linear
combination
EE 505 B Fall 2011
Assignment 8 Solutions
For full credit, you must show all of the steps or reasoning that you used to get the
answer.
1. (a) The denition of the region can be written as
B = cfw_( x1 , x2 ) : x1 > 0, x1 < x2 < x1 .
The region is shown i
EE 505 B Fall 2011
Assignment 8
For full credit, you must show all of the steps or reasoning that you used to get the
answer.
1. A joint probability density function is given as
c e x1
x1 > 0 and | x2 | < x1
0
f X1 X2 ( x 1 , x 2 ) =
otherwise
(a) Draw th
EE 505 B Fall 2011
Assignment 7 Solutions
1. (a)
Ecfw_ X =
=
x f X ( x ) dx
b
a
x
dx
ba
b
=
=
=
=
x2
1
ba 2 a
1
( b2 a2 )
2( b a )
(b a)(b + a)
2( b a )
b+2
2
(b) We use the direct method to nd the probability density function for Y:
f Y (y) =
i
f X (x)
EE 505 B Fall 2011
Assignment 7
For full credit, you must show all of the steps or reasoning that you used to get the
answer.
1. Let X be a random variable with a uniform probability density function.
f X (x) =
1
b a
0<a<x<b
0
otherwise
(a) Find Ecfw_ X .
EE 505 C, Autumn, 2010
Vector Calculus
1
1 Calculus Review
1.1 Integration over a plane (double integrals)
Integration over a plane is a special case of integrating over a surface, in which the
surface is a plane. The integral is
f ( x, y) dA
D
where D is
EE 505 B, Autumn, 2011
Linear Systems
1
1 Linear Systems
Denition 1. A linear system is a mapping of inputs to outputs that satisfy the principle
of superposition.
v( x )
v[m]
w( x )
w[m]
L
Figure 1: Block diagram of a linear system.
L must satisfy
1. Lcf
EE 505 B, Autumn, 2011
1
Linear Algebra
1
Introduction
The back cover of Gilbert Strangs book Introduction to Linear Algebra summarizes
all of linear algebra:
Ax = b
(N N)
Linear systems
Ax = b
(M N)
(N N)
(M N)
Least squares
Ax = x
Av = u
Eigenvalues
Sin