Download Probability and random variables

5-1-2011
Stochastic Hydrology
Probability and
random variables
Marc F.P. Bierkens
Professor of Hydrology
Faculty of Geosciences
Random variable: definition
A variable that can have a set of different values
generated by some probabilistic mechanism.
We do not know the value of a stochastic variable,
but we do know the probability with which a certain
value can occur.
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Example: throwing 2 dice
D
Pr(d)
2
1/36
3
2/36
4
3/36
5
4/36
6
5/36
2
3
4
5
6
7
6/36
8
5/36
9
4/36
10
3/36
11
2/36
12
1/36
7
8
9
10
11
12
0.18
0.16
0.14
probability
0.12
0.10
0.08
0.06
0.04
0.02
0.00
outcome
Expected value or mean
Nd
E[ D ]   d i Pr[ d i ]  2  1 / 36  3  2 / 36  .....  12  1 / 36  7
i 1
Estimated as:
1 n
Ê[ D ]   d j
n j 1
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Variance
Nd
VAR[ D]  E[( D  E[ D ]) 2 ]   (d i  E[ D]) 2 Pr[d i ]
i 1
 (2  7) 2  1 / 36  (3  7) 2  2 / 36  .....  (12  7) 2  1 / 36
 5.8333
Estimated as:
VÂR[ D ] 
1 n
 (di  Ê[D])2
n  1 i 1
Continuous variables
Histogram (Probability mass function) -> probability density function
fz(z)
Pr[Z=z1] = fz(z1)
WRONG!
z1
z
Pdf = probability mass per unit z
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Continuous variables
Pdf = probability mass per unit z
fz(z)
Pr[ z1  Z  z2 ] 
z2
f
Z
( z )dz
z1
z1
z2
z
Continuous variables
fz(z)
Pr[ z  Z  z  dz ]
dz
z
Formal definition probability density:
Pr[ z  Z  z  dz ]
f Z ( z )  lim
dz  0
dz

where
f
Z
( z )dz  1

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Continuous variables
Cumulative probability distribution function
FZ (z )
1
Pr[Z  z1 ]
0
z1
z
FZ ( z )  Pr[Z  z ]
Continuous variables
z
FZ ( z )  Pr[Z  z ] 
f
Z
( z )dz

f Z ( z) 
dFZ ( z )
dz
pdf
FZ (z )
fz(z)
cpdf
1
Pr[Z  z1 ]
0
z1
z
z1
z
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Continuous variables
z2
Pr[ z1  Z  z2 ]   f Z ( z )dz
z1
Pr[ z1  Z  z2 ]  FZ ( z2 )  FZ ( z1 )
pdf
cpdf
FZ (z )
fz(z)
1
0
z1
z2
z
z1
z2
z
Exercise
Consider the following probability density function:
fZ ( z) 
1  z /10
e
10
z0
1) Derive the cumulative probability distribution function.
2) What is the probability that Z lies between 5 and 10?
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Probability
Objectivistic definitions
• Classical
P( A) 
NA
All outcomes resulting in A

N Total number of possible outcomes
Example 2 dice : P(d  6) 
5 (5  1,4  2,3  3,2  4,1  5)
36
• Frequentistic
P( A)  lim
n 
nA number of trials resulting in A

n
Total number of trials
Probability
Objectivistic definitions
• Axiomatic (Kolmogorov, 1933)
1.
The probability of an event A is a positive number assigned to
this event:
P( A)  0
2.
The probability of the certain event (the event is equal to all
possible outcomes) equals 1:
3.
P(S )  1
If the events A and B are mutually exclusive then their union
equals the sum of the individual probabilities:
P( A  B )  P( A)  P ( B)
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Probability
Objectivistic definitions
• Axiomatic (Kolmogorov, 1933)
Exclusive events
Non-exclusive events
B
A
P ( A) 
Area A
Area S
S
A
B
S
Probability
Subjectivistic definitions
• Probability measures our “confidence” about the value or a range of values of a
property whose value is unknown.
• The probability distribution thus reflects our uncertainty about the unknown but
true value of a property.
Example 1: How tall is Marc Bierkens ?
Example 2: What is the IQ of George Bush?
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Measures of probability distributions
• Mean or Expected value (measure of locality)
Nd
E[ D ]   d i Pr[ d i ]
(discrete, e.g. throwing dice)
i 1

 Z  E[Z ]   z f Z ( z )dz

Estimated from data as:
(continuous: sum becomes an
integral and histogram a pdf)
̂ z 
1 n
 zi
n i 1
Measures of probability distributions
• Variance (measure of spread)

 Z2  E[(Z   Z ) 2 ]   ( z   Z ) 2 f Z ( z )dz

Estimated from data as:
ˆ z2 
1 n
 ( zi  ˆ Z )2
n  1 i 1
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Measures of probability distributions
• Skewness (measure of form)

CSZ 
E[(Z   Z )3 ]
 Z3

 (z  

Z
)3 f Z ( z )dz
 Z3
Estimated from data as:
1 n
 ( zi  ˆ z )3
n  1 i 1
ˆ
CS Z 
ˆ z3
Measures of probability distributions
• Rules with expected value and variance:
E[ a  bZ ]  a  b E[ Z ]
VAR[a  bZ ]  b 2 VAR[Z ]
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Examples of probability density functions
Probability density functions
Gaussian (normal)
normal) probability density:
density:
fZ ( z) 
1
2 Z
e
1  Z  Z 
 

2 Z 
2
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Relation between normal and lognormal pdf
Y  ln Z
Z lognormal distribution
Y normal distribution
Z  e
 Z2
Y  Y2 / 2
2 Y  Y2  Y2
e
(e  1)
Y  ln  Z 
 Y2  ln(
 Y2

 1)

2
Z
2
Z
2
Exercises
H ydraulic conductivity at som e unobserved location is m odelled w ith a log-norm al
distribution. T he m ean of Y=lnK is 2.0 and the variance is 1.5. C alculate the m ean and the
variance of K ?
H ydraulic conductivity for an aquifer has a lognorm al distribution w ith m ean 10 m /d and
variance 200 m 2 /d 2 . W hat is the probability that at a non-observed location the
conductivity is larger than 30 m /d?
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Two or more random variables
f ZY ( z , y )
0.0012
0.001
0.0008
0.0006
0.0004
0.0002
0
-100
-50
Bivariate pdf
y0
50
100
Pr[ z1  Z  z2  y1  Y  y2 ] 
20
40
-20
0z
-40
y2 z2
 f
ZY
( z , y )dzdy
y1 z1
Pr[ z1  Z  z2  y1  Y  y2 ]
dzdy
dy  0
Formal definition: f ZY ( z , y )  dzlim
0
Two or more random variables
FZY ( z , y )
Bivariate cpdf
1
0.8
0.6
0.4
0.2
0
-100
-50
y0
-20
- 50
- 100
- 40
- 20
-40
0z
FZY ( z , y )  Pr[ Z  z  Y  y ]
y
FZY ( z, y ) 
z
 f
 
ZY
( z , y )dzdy
f ZY ( z , y ) 
 2 FZY ( z , y )
zy
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Two or more random variables

Marginal probability density:
fZ ( z) 
f
ZY
( z , y )dy

Conditional probability:
FZ |Y ( z | y )  Pr{Z  z | Y  y )
Conditional pdf
f Z |Y ( z | y ) 
Independence of Z and Y
f ZY ( z , y )  f Z ( z ) fY ( y )
dFZ |Y ( z | y )
dz
Two or more random variables
Bayes’
Bayes’ theorem:
f Z |Y ( z | y ) 
fY | Z ( y | z ) f Z ( z )

f
Y |Z
( y | z ) f Z ( z )dz

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Two or more random variables
Covariance:
COV[ Z , Y ]  E[( Z   Z )(Y  Y )] 
 
  (z  
Z
)( y  Y ) f ZY (z , y )dzdy
 
Correlation:
 ZT 
COV[ Z , Y ]
 Z Y
In case of independence: COV[ Z , Y ]  0,  ZT  0
Two or more random variables
Properties of variance and covariance:
VAR[aZ  bY ]  a 2 VAR[ Z ]  b 2 VAR[Y ]  2ab COV[ Z , Y ]
VAR[aZ  bY ]  a 2 VAR[ Z ]  b 2 VAR[Y ]  2ab COV[ Z , Y ]
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Two or more random variables
Bivariate Gaussian probability distribution:
f ZY ( z , y ) 
1
2
2 Z  Y 1   ZY

 
  Z   Z
1
exp  
  
2
  2(1   ZY
)    Z


2
  Z  Y
  
  Y
2

 Z  Z
  2  

 Z
 Z  Y

  Y
 


 
Two or more random variables
Bivariate Gaussian probability distribution:
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Two or more random variables
Multivariate Gaussian probability distribution:
 Z1 
 
Z 
z 2
 
Z 
 N
 1 
 
 
μ 2
 
 
 N
f Z1 ...Z N ( z1 ,..., z N ) 
  12
 1 2 12

  2 1 21
Czz  

  
 N 1 N1
 1 N 1N 
 N2






 12 (z μ)T Czz1 (z μ)
1
e
(2 ) N / 2 | C zz |1/2
Appendix:
Elementary probability theory
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Probability Rules
A1
A2
Ai
Mutually exclusive (no intersection)
and exhaustive (filling all of S)
events Ai:
AM
S
M
 P( A )  P(S )  1
i 1
i
Probability Rules
{A  B}
intersection
A
B
{A  B}
Union
S
P ( A  B )  P( A)  P( B )  P( A  B)
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Probability Rules
{A  B}
Conditional probability of A given B:
B
A
P( A | B) 
{A  B}
P( A  B)
P( B)
S
P ( A  B )  P ( A | B ) P ( B )  P ( B | A) P ( A)
Probability Rules
{A  B}
Two events A and B are independent if:
A
B
P ( A  B )  P ( A) P ( B )
{A  B}
S
Because:
P ( A  B )  P ( A | B ) P ( B )  P ( B | A) P ( A)
The following also holds if A and B are independent:
P ( A | B )  P ( A)
P ( B | A)  P ( B )
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Probability Rules
A1 A2
Total probability theorem:
Ai
M
M
i 1
i 1
P ( B )   P ( Ai  B )  P ( B | Ai ) P ( Ai )
AM
{Ai  B}
B
S
Bayes’ Theorem
P( Ai | B) 
P( B | Ai ) P( Ai )
M
 P( B | A ) P( A )
j 1
j
j
Used for updating prior probability P(Ai)
given observations B and likelihood P(B|Ai)
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