CHAPTER 6
CORRELATION AND SIMPLE
LINEAR REGRESSION
CORRELATION
quantifies
the strength of the linear relationship
between x and y
Noted as
The range of the correlation coefficient is from -1 to
+1
population correlation coefficient is
The value shows how
1
ECE 2221
SIGNALS AND SYSTEMS
Fourier Series
ECE 2221 Signals and Systems
Motivation for using Fourier series
2
Electrical engineers instinctively think of signals in terms of
their frequency spectra and think of system in terms of their
frequency respo
Electrostatics
=
=0
=
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=0
=
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=
Magnetostatics
Study of electric and magnetic phenomena under static
conditions are known as electrostatics and magnetostatics.
In 800 B.C., the Greeks discovered magnetite, certain kind of
stones which exhibits
Electrostatics
=
=0
=
=+
=0
=
=0
=0
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Magnetostatics
Michael Faradays hypothesis
if a current can produce a magnetic field, a magnetic field should
produce a current in a wire
In 1831, through numerous experiments, Faraday (and also Joseph Henry)
disc
NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL
EQUATIONS
order solve theinitialvalue problem of the form
dy
=f ( x , y ) ,two methods will be
dx
discussed ; Euler methodRunge Kuttamethod .
Eulers method
Some of the ODE cannot be solved analytically. This i
CHAPTER 7: SOLUTION OF LINEAR SYSTEM OF EQUATIONS
GAUSS ELIMINATION
Deals with simultaneous linear algebraic equations that can be represented generally
as
a11 x 1+ a12 x 2 + a1 n x n=b 1
a21 x 1 +a 22 x2 + + a2 n xn =b2
an 1 x1 +a n 2 x 2+ +a nn x n=b n
Magnetization : giving magnetic property; to make magnetic
Done by applying external magnetic field to a material.
The degree of magnetization of a material depends on the
magnetization vector of the material, which is the vector sum
of the magnetic dipol
Gauss law for magnetism
Ampere circuital law
The number of magnetic field lines through a surface, denoted as , a
scalar quantity given the unit of Weber (Wb) or tesla-meter squared
(Tm2)
The concept is as the same as electric flux.
Mathematically defined
INTERPOLATION
Many times, data is given only at discrete points such as
xn , yn
So, how then does one find the value of
continuous function
through the
x
n 1
f x
y
x0 , y0 , x1 , y1
, .,
at any other value of
may be used to represent the
n 1
data valu
NUMERICAL INTEGRATION AND DIFFERENTIATION
Newton Cotes Integration Formulas
Newton cotes integration formulas are the most common numerical integration schemes. They are
based on the strategy of replacing a complicated function or tabulated data with an a
Chapter 2
SEMICONDUCTOR MATERIALS AND DIODES
ECE 1312 ELECTRONICS
1
Intrinsic Semiconductors
Intrinsic Semiconductors
DEFINITION:
A pure form of semiconductor crystal is called intrinsic semiconductor.
The atoms have the four valance electrons are in gro
CHAPTER 4
Hypothesis Testing
Recap
In the previous chapter we illustrated how to construct a
confidence interval estimate of a parameter from a sample
data. However, many problems in engineering require that we
decide whether to accept or to reject a sta
CHAPTER 5
ANOVA
What is ANOVA ?
is a hypothesis-testing technique used to test the
equality of three or more population means by
examining the variances of samples that are
taken.
ANOVA allows one to determine whether the
differences between the samples
CHAPTER 2a
DISCRETE RANDOM VARIABLE AND ITS
PROBABILITY DISTRIBUTION FUNCTIONS
Term with definitions
Random
variable assigns a real number to each outcome in the sample space
of a random experiment. It involves random process and the process will map the
CHAPTER 2b
CONTINUOUS RANDOM VARIABLE AND ITS
PROBABILITY DISTRIBUTION FUNCTIONS
Continuous probability
distribution
Probability density function - function that describes the relative
likelihood for the random variable to take on a given value.
The pro
SHF1124
CHAPTER 3: INTODUCTION TO STATISTICS
The Statistical Process
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Plan the Investigation:
What? How? Who? Where?
Collect the Sample
3.1 Introduction
CHAPTER 1
PROBABILITY
What is probability?
Chance that something will happen
PROBABILITY = CHANCE
Terms with definitions
Random experiment an experiment that can result in different outcomes, even though it
repeated in the same manner every time.
Sampl
Transform of direvativas
Theorem.
Example 1.
Then solution of initial-value problem is
Example 2.
Hence a solution of given equation
Theorem.
Example 1.
Theorem.
Example 1.
LAPLACE TRANSFORMATION.
In elementary calculus you learned that differentiation and integration are
transforms; this means, that these operations transform a function into another
function. For example, the function is transformed, in turn, into a linear
Power series, solution about ordinary points.
If L<1 then the power series converges at x. Radius of convergence: R=lim|cn+1/cn| as n tents to
infinity.
Suppouse the linear second order differential equation
'
'
a2 (x ) y + a1 ( x) y + a0 (x) y=0 (5)
y '
Solution about singular points.
Definition.
Theorem.
Example 1. Use Frobenius method to solve given differential equation about a regular
singular point
'
'
x=0, 3 x y + y y =0 .
Solution. We will try to find a solution of the form
Then
So
Then
And
y=C 1
MODELING WITH HIGHER-ORDER
DIFFERENTIAL EQUATIONS.
1. SPRING/MASS SYSTEMS:
FREE UNDAMPED MOTION
HOOKES LAW Suppose that a flexible spring is suspended vertically from a
rigid support and then a mass m is attached to its free end. The amount of stretch,
or
Cauchy-Euler equation
Definition .
A linear differential equation of the form
an x n y (n) +an1 x n1 y (n1) + a1 x y ' + a0 y=g ( x )
an , an1 , , a0
where the coefficients
are constants, is called a
Cauchy-Euler
equation.
We start the discussion with a d
Variation of parameters
Let given a linear second-order differential equation
'
'
a2 (x ) y + a1 ( x) y + a0 (x) y=g (x)(1)
For the linear second-order differential equation
solution in the form
(1) we seek a particular
y p=u1 ( x ) y 1 +u2 ( x ) y 2
wher
Reduction of order
1. In case equation on form
(n)
=f (x)
2. In case the dependent variables
that is
solved by n times integrating.
is missing from our differential equation,
x , ' , ) =0
, we make the substitution
F
'=
. This entails
= p ' .
The diffe
a
Homogeneous linear equations with constant coefficients
Definition. DE
(n )
an y + an1 y
(n1)
'
'
+a 2 y + a1 y +a 0 y=0(1)
is said to be norder homogeneous linear differential equation with constant
coefficients if the coefficients
ai , i=1,2, ,n
are r
Linear equations
Definition 1.
(linear equation)
A first-order differential equation of the form
a1 ( x )
dy
+a ( x ) y=g ( x ) (1)
dx 0
is said to be a linear equation in the dependent variable
When
g ( x ) =0
y
.
, the linear equation (1) is said to be
Linear transformations
Definition.
Let
V
and W be vector spaces.
L
A linear transformation
L(u)
vector
in
W
to each u
a)
L ( u+ v )=L ( u ) + L(v)
b)
L ( ku )=kL ( u )
V =W
If
of
V
in
V
u V
, the linear transformation
is a function assigning a unique
such
Homogeneous equations
If a function f
Definition 1.
possesses the property
f ( tx , ty ) =t f ( x , y)
for some real number , the f
is said to
be homogeneous function of degree .
For example,
f ( x , y ) =x3 + y 3
is homogeneous function of degree 3 ,
f (
Differential equations
Definition 1. (Differential equation)
An equation containing the derivatives of one or more depended variables,
with respect to one or more independent variables, is said to be a differential
equation.
Definition 2. If an equation c