# Ee315 Unit 5 Work Sheet

Satisfactory Essays
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KING FAHD UNIVERSITY OF PETROLEUM AND MINERALS ACHB/CE
EE315: PROBABILISTIC METHODS IN ELECTRICAL ENGINEERING
ELECTRICAL ENGINEERING DEPARTMENT
Student ID
Student Name
Section
1
MAJOR II EXAMINATION
This is a secure exam. All KFUPM/ACHB examination regulations apply.
Course:
EE315: PROBABILISTIC METHODS IN ELECTRICAL ENGINEERING
Semester-year:
131-2013-2014
Day-Date:
THURSDAY November 14, 2013
Start time:
07:30 AM
Finish time:
09:00 AM
Instructor Name
Dr. Belloui Bouzid
Signature
General Information to Students:
Answer the questions on these pages.
Hand in any separate work sheets with these pages.
QUESTION NO.
MARKS
MARKS AWARDED 1
30
2
40
3
30
Totals:
100
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KING FAHD UNIVERSITY
Density Function
( ) ( )
( ) ∫ ( )
( ) ( ) ∫ ( )
( ) { } ∫ ( )
b) Summarize the importance of Gaussian random variable
The Gaussian density enters in all areas of science and technology.
Accurate description of many practical and significant quantities which are the results of small independent random effects.
The importance stems from its accurate description of many practical and significant real-world quantities. c) Draw the Gaussian density function ( )
( )

( )
Where and are real constants. Its maximum value

occurs at . Its spread about the point . The function decreases to 0.607 times its maximum at and
. It was first derived at ( )

√ at ( )

( ) at ( )

√ f (x) X
X X a
-2 -1 -0.50 0.5 1 2
0.5
x
0
0.1 0.1
0.2
0.3
0.2
0.1
5
5
12/9
KING FAHD UNIVERSITY OF PETROLEUM AND MINERALS ACHB/CE
EE315: PROBABILISTIC METHODS IN ELECTRICAL ENGINEERING
b) M g ( x) x
2 0

x 
0 x
2 0

0 x 
( a ) b For
(
)
(
)
( ) [ (
)]
( )
(
)
For
(
)
( )
( ) (
)
( )
( )
( )
{
(
)
( )
( )
{
∫ ( )
(
)
∫ ( )
( )
( )
4
4
4
4
13/9
KING FAHD UNIVERSITY OF PETROLEUM AND MINERALS ACHB/CE
EE315: PROBABILISTIC METHODS IN ELECTRICAL ENGINEERING

2
G (x) forb  x
0 x
0.5
0 x
1
(b)
2 0

x 
2 0

x 
4
c)
Binomial, Poisson (discrete RV), Uniform, Exponential, Rayleigh (Continuous RV)
2
2
2
2
2
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KING FAHD UNIVERSITY OF PETROLEUM AND MINERALS ACHB/CE
EE315: PROBABILISTIC METHODS IN ELECTRICAL ENGINEERING
QUESTION 3
a) [ ] ∫ ( ) ∫
∫ ∫
∫ ([ ] ∫ ) ( [ ] )
= ( )
2
2
2
4
b) ( ) ( )
[ ( )] ∫ ( ) ( ) ( ) ( ) ( ) [ ( )] ∫( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ( )] ∫( ) ( ) ( ) ∫ ( ) ( ) ( ) ∫ ( )
∫ ∫
∫ ([ ]( ) ∫ ( ) ) ( ) [ ]
2
2
2
2
2
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KING FAHD UNIVERSITY OF PETROLEUM AND MINERALS ACHB/CE
EE315: PROBABILISTIC METHODS IN ELECTRICAL ENGINEERING
c)
[( ) ] ∫( ) ( )
By expanding the previous function we have the following:
[( ) ] [ ] [ ] [ ]
Put [ ]
The equation becomes the following
[( ) ] [ ] [ ]
We have [ ] [ ]
And
At the end: [( ) ] [ ]
2
2
2
2