Ee315 Unit 5 Work Sheet

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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:

Read and follow the instructions for each question carefully.

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

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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

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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