1819SEM1-CE2407.pdf - NUS CONFIDENTIAL CE2407 NATIONAL...

NUS CONFIDENTIAL
CE2407
NATIONAL UNIVERSITY OF SINGAPORE
______________________________________________________________________
INSTRUCTIONS TO CANDIDATES
1. Please write your student number only. Do not write your name.
2. This assessment paper contains FOUR(4) questions and comprises NINETEEN(19)
printed pages.
3. Answer ALL questions. All questions carry equal marks.
4. Please start each question on a new page.
5. This is an “ CLOSED BOOK ” assessment.
6. Programmable or Graphical calculator is NOT allowed for this examination.
(eg. Electronic calculator is permitted for this exam).
CE2407 – ENGINEERING & UNCERTAINTY ANALYSES
(Semester 1: AY2018/2019)
Time Allowed: 2.5 Hours

NUS CONFIDENTIAL
- 2 - CE2407
Useful General Formulae for CE2407

NUS CONFIDENTIAL
- 3 - CE2407
߲ݑ
߲ݔ
ൎ
ݑ
௜ାଵ,௝
െ ݑ
௜,௝
ሺ∆ݔሻ
߲ݑ
߲ݕ
ൎ
ݑ
௜,௝ାଵ
െ ݑ
௜,௝
ሺ∆ݕሻ
Forward-difference approximation
߲
ଶ
ݑ
߲ݔ ଶ
ൎ
ݑ
௜ାଶ,௝
െ 2ݑ
௜ାଵ,௝
൅ ݑ
௜,௝
ሺ∆ݔሻ ଶ
߲
ଶ
ݑ
߲ݕ
ଶ
ൎ
ݑ
௜,௝ାଶ
െ 2ݑ
௜,௝ାଵ
൅ ݑ
௜,௝
ሺ∆ݕሻ ଶ
߲ݑ
߲ݔ
ൎ
ݑ
௜,௝
െ ݑ
௜ିଵ,௝
ሺ∆ݔሻ
߲ݑ
߲ݕ
ൎ
ݑ
௜,௝
െ ݑ
௜,௝ିଵ
ሺ∆ݕሻ
Backward-difference approximation
߲
ଶ
ݑ
߲ݔ
ଶ
ൎ
ݑ
௜,௝
െ 2ݑ
௜ିଵ,௝
൅ ݑ
௜ିଶ,௝
ሺ∆ݔሻ ଶ
߲
ଶ
ݑ
߲ݕ ଶ
ൎ
ݑ
௜,௝
െ 2ݑ
௜,௝ିଵ
൅ ݑ
௜,௝ିଶ
ሺ∆ݕሻ
ଶ
߲ݑ
߲ݔ
ൎ
ݑ
௜ାଵ,௝
െ ݑ
௜ିଵ,௝
2ሺ∆ݔሻ
߲ݑ
߲ݕ
ൎ
ݑ
௜,௝ାଵ
െ ݑ
௜,௝ିଵ
2ሺ∆ݕሻ
Centred-difference approximation
߲
ଶ
ݑ
߲ݔ ଶ
ൎ
ݑ
௜ାଵ,௝
െ 2ݑ
௜,௝
൅ ݑ
௜ିଵ,௝
ሺ∆ݔሻ ଶ
߲
ଶ
ݑ
߲ݕ ଶ
ൎ
ݑ
௜,௝ାଵ
െ 2ݑ
௜,௝
൅ ݑ
௜,௝ିଵ
ሺ∆ݕሻ ଶ

NUS CONFIDENTIAL
- 4 - CE2407
Note: Before checking this Table, round-off the variable z into only two decimals, e.g.,
0.548~0.55, 0.234~0.23, 0.765~0.77).

NUS CONFIDENTIAL
- 5 - CE2407

NUS CONFIDENTIAL
- 6 - CE2407

NUS CONFIDENTIAL
- 7 - CE2407
Lecture 2 Probability (1)
Mutually exclusive
2 events
) P(E ) P(E ) E P(E
2 1 2 1
  
3 events
1 2 3 1 2 3
P(E E E ) P(E ) P(E ) P(E )     
Not mutually exclusive
1 2 1 2 1 2
P(E E ) P(E ) P(E ) P(E E )     
1 2 3 1 2 3 1 2 2 3 1 3 1 2 3
P(E E E ) P(E ) P(E ) P(E ) P(E E ) P(E E ) P(E E ) P(E E E )         
Operational rules
Commutative A  B = B  A, AB = BA
Associative (A  B)  C = A  (B  C) (A B)C = A(BC)
Distributive (A  B) C = AC  BC, (AB)  C = (A  C) (B  C)
De Morgan’s rule
1 2 n 1 2 n
E E E E E E        K K
,
1 2 n 1 2 n
E E E E E E     K K
Lecture 3 Probability (2)
P(E
1
|E
2
) = P(E
1
E
2
)/P(E
2
)
1 2 1 2
P(E E ) 1 P(E E )  
, but
1 2 1 2
P(E E ) 1 P(E E )  
P(E
1
E
2
E
3
) = P(E
1
|E
2
E
3
)  P(E
2
|E
3
)  P(E
3
)
Statistically independent
P(E
1
|E
2
) = P(E
1
), P(E
1
E
2
) = P(E
1
)  P(E
2
), P(E
1
E
2
...E
n
) = P(E
1
) P(E
2
)...P(E
n
)
Theorem of total probability P(A) = P(A|E
1
)P(E
1
) + P(A|E
2
)P(E
2
) + … + P(A|E
n
)P(E
n
)
Bayes theorem P(E
1
|A) = P(A|E
1
)P(E
1
)/P(A)
1 1 2 2
( | ) ( )
( | )
( | ) ( ) ( | ) ( ) ... ( | ) ( )
i i
i
n n
P A E P E
P E A
P A E P E P A E P E P A E P E

  
Lecture 4 Random variables
Continuous ( ) ( ) for all
X
F x P X x x  
Discrete
( ) ( ) ( )
i
i
X i i X
all x x
F x P X x p
x

  

( )
( ) X
X
dF x
f x
dx

,
( ) ( )
x
X X
F x f x dx

 


( ) ( ) ( )
X X
P a X b F b F a    
 
1
[ ] ( ) or ( )
N
X i X i X
i
E X x p x x f x dx 



 



NUS CONFIDENTIAL
- 8 - CE2407
 
1
[ ( )] ( ) ( ) ( ) ( )
N
i X i X
i
E g X g x p x or g x f x dx






1
[ ] ( ) or ( )
N
n n n
i X i X
i
E X x p x x f x dx



 

 


2 2 2 2
1
[( ) ] ( ) ( ) or ( ) ( )
N
X X i X X i X X
i
E X x p x x f x dx    



    


2 2 2 [ ]
X X
E X    
, Coefficient of variation
X
X
X




,
Skewness
3
3
3 3
[( ) ] 1
( ) ( ) X
X X
X X
E X
x f x dx


 



  

Lecture 5 Distribution (continuous)
Normal distribution
2
1 1
( ) exp ( , )
2
2
X
x
f x N x

 

 
 

 
       
 
 
 
 
 
2 1
(0,1) ( ) exp
2
2
x
N x 

 
  
 
 
( ) ( ) ( )
s
S
s F s x dx 

  

( ) 1 ( ) 1
p p
s s p       
When p < 0.5,
1 (1 )
p
s p    
Lognormal distribution
-4 -3 -2 -1 0 1 2 3 4
0.0
0.1
0.2
0.3
0.4
0.5
N(0,1)
area = 1-p
probability = p
s
p
s
f
S
(s)
2
1 1 ln
( ) exp ( , ) 0
2
2
X
x
f x LN x
x

 

 
 
  
     
 
 
   
 
Parameters:  = E(ln X ), 
2
= var(ln X )
2
[ ] exp( 0.5 )
X
E X      
, 2
ln 0.5
X
    
 
2
2 2 2
2
ln 1 ln 1 (for 0.3)
X
X X X
X

   

 
     
 
 
ln
( )
X
a
F a


  
 
 
 