CE/EE 302:
Probabilistic Methods in Electrical and Computer Engineering
Chapter 2  Week 2
Dr. Pierre E. ABICHAR & Dr. Ahmed Youssef
1
Outline
Deterministic model
Random experiment
Sample space
Event
Set Theory
Relative frequency
Probability
2
Why
CE/EE 302:
Probabilistic Methods in Electrical and Computer Engineering
Chapter 2  Week 4
Dr. Pierre E. ABICHAR & Dr. Ahmed Youssef
1
Chapter Outline
1)
2)
3)
4)
5)
Conditional Probability
Independence of Events
Sequential Experiments
Binomial Probabili
CE/EE 302:
Probabilistic Methods in Electrical and Computer Engineering
Chapter 2  Week 3
Dr. Pierre E. ABICHAR & Dr. Ahmed Youssef
Chapter Outline
1)
2)
3)
4)
Introduction
Specifying Random Experiments
Events & Sample spaces
Computing Probabilities Usi
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CONSOLIDATED STATEMENT OF FINANCIAL POSITION Boubyan Bank
As at 31 December 2014
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Assets
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International Journal of Lean Six Sigma
A Six Sigma and DMAIC application for the reduction of defects in a rubber gloves
manufacturing process
Ploytip Jirasukprasert Jose Arturo GarzaReyes Vikas Kumar Ming K. Lim
Article information:
Downloaded by Kuwai
Preparation for the Quiz
Problem 1. Evaluate the derivative of the following functions
i. y = esin(3x)
ii. y = sin(sin(3x)
iii. y = tan(ln(2x) + x)
Problem 2. Find y 0 in the following implicit functions
i. y cos(y) + x = 3
ii. e y + y 2x = sin(x)
iii. (1
Differential Equations
Preparation
The following problems are the minimum knowledge we need for an elementary course in Ordinary
Dierential Equations. The materials can be found in any basic course in Calculus.
Problem 1. Find a such that the following fu
Problem 1. Draw some typical solutions of the following equation
y = y(4 y).
Solution.
1. Find equilibrium. We put y = 0 in the above equation to obtain y = 0 and y 4 as
equilibrium.
2. Do stability analysis. We have the following digram
Unstable
Stable
3
Preparation for the FinalI
Niksirat
No mark is ocially assigned to the scripts.
Problem 1. Find the Laplace tarnsform of the following functions using the deinition
i. f (t) = u(t)
ii. f (t) = t
iii. f (t) = e2t
iv. f (t) =
1 0<t<1
0 t>1
Problem 2. Find
Preparation for the Midterm III
part I
The due date of collecting this homework is 3rd Dec. I DO NOT officially assign any
mark to the scripts, nevertheless, I look into your solutions and consider your works.
Problem 1. Find two linearly independent solu
Integration Techniques
Find the following integrals:
Z
Z
Z
dx
xln(x)
tan(x)
dx
ln(cos(x)
cos(x)
dx
sin2x + 2 sinx + 2
Z
dx
x(ln2x + 4lnx + 5)
Z
lnx
dx
x2
Z
ex cos(x) dx
Z
Z
tan4x sec4xdx
Z
sin1xdx
Z p
9 4x2 dx
2x 1
dx
x2 2x + 10
1
Preparation for the Midterm III
part II
No mark is ocially assigned to the handed scripts, nevertheless, I look into your solutions
and consider your works.
Problem 1. Consider the equation
(1 + x2)y 00 2xy 0 + 2y = 0:
i. Verify that the function y = x is
Introduction to Differential Equations
Dr. Antonios Kalampakas
MAT300  MA 266
1 / 18
Definitions
Differential Equations: An equation containing the derivatives of an
unknown function (called dependent variable) with respect to one or
more independent var
Problems Separable Differential Equations
Dr. Antonios Kalampakas
Differential Equations
1 / 10
Example
Example. Solve the following DE
dy
= sin5x
dx
2 / 10
Example
Example. Solve the following DE
dx + e 3x dy = 0
3 / 10
Example
Example. Solve the followi
InitialValue Problems
Dr. Antonios Kalampakas
Ordinary Differential Equations  MA 266
1 / 10
InitialValue Problems
Definition The problem of solving an nthorder ODE subject to n
conditions specified at x0 is
dny
dx n
= f (x, y , y 0 , . . . , y (n1) )
Exact Differential Equations
Dr. Antonios Kalampakas
Differential Equations
1 / 18
Exact Differential Equation
If f (x, y ) is a function of two variables with continuous first partial
derivatives in a region R of the xyplane, then its differential is
d
Reduction of Order
Dr. Antonios Kalampakas
Differential Equations
1/8
Reduction of Order
Given a solution y1 (x) 6= 0 of the second order linear differential
equation
y 00 + P(x)y 0 + Q(x)y = 0
we obtain a second linearly independent solution by
Z
y2 = y1
Exercises on Direction Fields
Dr. Antonios Kalampakas
Differential Equations
1/6
Example 1
Find which one of the following DEs corresponds to the direction field
A.
dy
= xy
dx
2/6
B.
dy
= x 2y 2
dx
C.
dy
= 1xy
dx
D.
dy
= x 2 y 2
dx
Example 2
Find which on
HigherOrder Differential Equations: Initial and Boundary
Value Problems
Dr. Antonios Kalampakas
Ordinary Differential Equations  MA 266
1 / 10
Initial Value Problem for Higher order DEs
Consider the following IVP
an (x)
d n1 y
dy
d ny
+
a
(x)
+ + a1 (x)
Direction Fields  Separable Equations
Dr. Antonios Kalampakas
Differential Equations
1 / 22
Direction Fields
Assume that y = y (x) is a solution of a firstorder differential
equation
dy
= f (x, y ).
dx
The function f (x, y ) is called the slope function
HigherOrder Differential Equations: Homogeneous
Equations
Dr. Antonios Kalampakas
Ordinary Differential Equations  MA 266
1 / 17
Homogeneous Equations
A linear nth order differential equation of the form
an (x)
d ny
d n1 y
dy
+ an1 (x) n1 + + a1 (x)
+
Linear FirstOrder Differential Equations
Dr. Antonios Kalampakas
Differential Equations
1 / 11
Definition
A firstorder linear differential equation has the form
a1 (x)
dy
+ a0 (x)y = g (x).
dx
To solve we divide both sides by a1 (x) to write it in the S
Partial Derivatives
Dr. Antonios Kalampakas
Differential Equations
1/3
Partial Derivatives
Partial Derivatives. For functions with two variables f (x, y ) we find
the partial derivative
f
x
or fx with respect to x
by assuming that y is constant and simila
applied calculus for business economics and the social and life sciences
MATH 175

Spring 2016
8. Find an equation of a line in pointslope
form and in slopeintercept form that satisfies the
following conditions. Graph parts (a), (b), (c) and (d).
a)
x intercept:
8
and
y
intercept: 6
P.inL * dope
h ( tro) p" (o ,6\
yv13
*.
Slo
ll('rro1
9" t.
applied calculus for business economics and the social and life sciences
MATH 175

Spring 2016
Running Head: APPLICATION OF DIFFERENTIATION
Research Project:
Course Name:
Course Code:
Project Tittle: Application of Differentiation in Economics: Revenue and Sale Maximization.
APPLICATION OF DIFFERENTIATION
Introduction
In real life, businesses are a
applied calculus for business economics and the social and life sciences
MATH 175

Spring 2016
11. Graph the following
quadratic functions.
Find the verrAy
ail the
@, I ;#;:
,
r
;:jlvH;
_
v i+^^,,
.,"0,",uf,r"*T;ffi,ipept,
and
b)h@)=_2,x2+4x_5
Po xinler.<pF
No \oLlrbn
* q ( z)
= (u)t
l'4 = 2q (O
\in[".cep[ (=o
(o)\ t(") .f,' = "'E
l(o,
pc
applied calculus for business economics and the social and life sciences
MATH 175

Spring 2016
lr
The line is vertical and passing through
d)
(7,1)
o $eco.rse 4 h" (ir,. ts
v",.
k'cJ
[lsl Y q x.iS *r,cfw_
ln l.t .p[ on Xcrrrts in (+ro\
4lon is,. c1u*lton h *=?
[ir,e
*T[.
P".ra'
,
elThe line passes through (1,7)
ancfw_pTlle\o
the line passing
applied calculus for business economics and the social and life sciences
MATH 175

Spring 2016
",
9' Suppose the demand for theatre tickets are 1000 tickets when the price
is at $5.00 per ticket
and 200 tickets when the price is at $15.00 per ticket. Find the
demand equation in terms ofp
(price) and q (quantity), assuming it is linear. Graph the de