Chapter 3 Analytical Models of Random Phenomenon

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Chapter 3 Analytical Models of Random Phenomenon
CIVL 181 Modelling Systems with Uncertainties

• Instructor Prof. Wilson Tang

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Random variables

• A device to
• formalize description of event
• facilitate computation of probability

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a) formalize description of event
Chapter 2 Chapter 3
Excessive settlement X gt 3 cm
Delay in project T gt 60 days
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b) facilitate computation of probability
introduce algebra and calculus
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Discrete random variables
Bulldozers example
GGG BBG GGB BGB GBG GBB BGG
BBB
X no. of good bulldozers
Assume P(G) P(B) ½ and s. i.
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PMF

• Complete information about X

• Compute probability of any event relating to X

e.g.
P(X gt 1) ?
3/8 1/8 1/2
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CDF
P(X lt x)
1
7/8
1/2
1/8
x
0
1
2
3
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Continuous random variables
Displacement
P( X3 ) ? ? 0 no area / width
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CDF
P( 3ltxlt4)
FX(4) FX(3)
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CDF
PX(x)
FX(x)
PMF
fX(x)
FX(x)
PDF
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Example
(a) Determine a
fX(x)
? area 1
3a
10a30a1
?a 0.025
a
x
20
5
10
15
X location of an accident
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(b) P( 15 lt x lt 20) ?
Method 1 by area
P( 15 lt x lt 20) 50.075 0.375
Method 2 by integration
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Method 3 by CDF
P( 15 lt x lt 20) FX(20) FX(15) 1-0.625
0.375
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Example
fX(x) 2 e-2x x gt 0
X snowfall in m
a)
P ( x lt 1)
Is this probability reasonable?
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fX(x)
fX(x) 2 e-2x x gt 0
2.0
0.27
x
1
0
P ( x lt 1)
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b)
Open door, find out 1 m snow, then P(no class) ?
P( X gt1.5 X gt1)
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Example
FX(x)
1.0
ln (x/3)
x
h
3
0
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a) h ?
ln ( h/3 ) 1
h 3e 8.15
b) P (4ltxlt5) ?
ln (5/3) ln (4/3) 0.511 - 0.288 0.223
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c) fX(x) ?
Check
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Main descriptors of R.V.

• The PMF or PDF completely define the r.v.
• Descriptors give partial information about the
  r.v.

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Mean value
Define ? E(X)
expected value of X or mean value of X
a measure of central tendency
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Example
E(X) 20.130.440.450.1 3.5
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Consider another PMF
E(X) 3.5
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Variance
Define Var (X)
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Standard deviation ?X
Coefficient of variation ? ?X/?X
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for continuous r.v.
centroid
Moment of inertia
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Example
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For a previous example
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Other measure of central tendency

• Median xm - 50 percentile value
• Mode x - most likely value

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Example
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Measure of spread

• Standard deviation ?X
• ?X dimensionless
  
• range

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Measure of skewness
or
gt 0 positive skewness lt 0 negative
skewness 0 symmetrical
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Expected value of function
recall
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Example delay
weeks
Penalty 2000x2
E penalty E 2000x2
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Lets try a short cut
E penalty
2000 E(x)2 2000(1.5)2 4500 ? 4667
Why?
Only an approximation
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Recall
After some algebra,
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Example total load on the roof
fX(x)
x
6
3
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0.64
mode 3
xm 3.79