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STATISTICS
Random Variables and
Probability Distributions
Professor Ke-Sheng Cheng
Department of Bioenvironmental Systems Engineering
National Taiwan University
Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University
Definition of random
variable (RV)

For a given probability space (  ,A, P[]),
a random variable, denoted by X or X(),
is a function with domain  and
counterdomain the real line. The function
X() must be such that the set Ar, denoted
by Ar   : X ()  r , belongs to A for
every real number r.
Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University
Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University
Cumulative distribution
function (CDF)

The cumulative distribution function of a
random variable X, denoted by FX (), is
defined to be
FX ( x)  P[ X  x]  P{ : X ()  x}
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x  R
Dept of Bioenvironmental Systems Engineering
National Taiwan University

Consider the experiment of tossing two
fair coins. Let random variable X denote
the number of heads. CDF of X is
x0
 0
0.25 0  x  1

FX ( x )  
0.75 1  x  2

2 x
 1
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FX ( x)  0.25I [ 0,1) ( x)  0.75I [1, 2) ( x)  I [ 2,  ) ( x)
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Indicator function or
indicator variable

Let  be any space with points  and
A any subset of . The indicator
function of A, denoted by I A () , is the
function with domain  and
counterdomain equal to the set
consisting of the two real numbers 0
and 1 defined by
1 if   A
I A ( )  
0 if   A
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Hydrology and Spatial Modeling
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Discrete random variables


A random variable X will be defined to be discrete
if the range of X is countable.
If X is a discrete random variable with values
x1 , x2 ,, xn ,, then the function denoted by
f X () and defined by
P[ X  x j ] if x  x j , j  1,2,, n,
f X ( x)  
0
if x  x j

is defined to be the discrete density function of X.
Lab for Remote Sensing
Hydrology and Spatial Modeling
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Continuous random variables


A random variable X will be defined to be
continuous if therex exists a function f X ()
such that FX ( x)   f X (u)du for every real
number x.
The function f X () is called the probability
density function of X.
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Hydrology and Spatial Modeling
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Properties of a CDF
FX ()  lim FX ( x)  0
x  
FX ()  lim FX ( x)  1
x  
FX (a)  FX (b) for a  b
FX () is continuous from the right, i.e.
lim F
0 h 0
X
( x  h )  FX ( x )
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Hydrology and Spatial Modeling
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Properties of a PDF
f X ( x)  0



x  R
f X ( x)  1
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Example 1

Determine which of the following are
valid distribution functions:
1  [e2 x / 2] x  0
FX ( x)  
2x
x0
 e /2
x
FX ( x)  u ( x  a)  u ( x  2a)
a
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1 x  0
u ( x)  
0 x  0
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Lab for Remote Sensing
Hydrology and Spatial Modeling
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Example 2

Determine the real constant a, for arbitrary
real constants m and 0 < b, such that
f X ( x)  ae
 x m / b
x  R
is a valid density function.
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



Function
f X (x) is symmetric about m.

f X ( x)dx  2 ae
m
 ( x  m) / b

dx  2ab e dy  2ab  1
y
0
a  1/ 2b
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Characterizing random
variables


Cumulative distribution function
Probability density function






Expectation (expected value)
Variance
Moments
Quantile
Median
Mode
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Expectation of a random
variable

The expectation (or mean, expected value) of
X, denoted by  X or E(X) , is defined by:
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Rules for expectation

Let X and Xi be random variables
and c be any real constant.
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Variance of a random
variable
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
 X  Var( X )  0
is called the
standard deviation of X.
 
Var[ X ]   X2  E[ X 2 ]  ( E[ X ]) 2  E X 2   X2
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Rules for variance
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

Two random variables are said to be
independent if knowledge of the value
assumed by one gives no clue to the value
assumed by the other.
Events A and B are defined to be
independent if and only if
P[ AB]  PA  B  PAPB
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Hydrology and Spatial Modeling
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Moments and central moments
of a random variable
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Properties of moments
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Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University
Quantile

The qth quantile of a random variable X,
denoted by  q , is defined as the smallest
number  satisfying FX ( )  q .
Lab for Remote Sensing
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Median and mode


The median of a random variable is the 0.5th
quantile, or  0.5 .
The mode of a random variable X is defined
as the value u at which f X (u ) is the
maximum of f X () .
Lab for Remote Sensing
Hydrology and Spatial Modeling
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Note: For a positively skewed distribution, the
mean will always be the highest estimate of
central tendency and the mode will always be
the Lowest estimate of central tendency
(assuming that the distribution has only one
mode). For negatively skewed distributions,
the mean will always be the lowest estimate of
central tendency and the mode will be the
highest estimate of central tendency. In any
skewed distribution (i.e., positive or negative)
the median will always fall in-between the
mean and the mode.Dept of Bioenvironmental Systems Engineering
Lab for Remote Sensing
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Moment generating function
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Usage of MGF


MGF can be used to express moments
in terms of PDF parameters and such
expressions can again be used to
express mean, variance, coefficient of
skewness, etc. in terms of PDF
parameters.
Random variables of the same MGF
are associated with the same type of
probability distribution.
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
The moment generating function of
a sum of independent random
variables is the product of the
moment generating functions of
individual random variables.
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Expected value of a function
of a random variable
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
If Y=g(X)

E[ g ( X )]   g ( x) f X ( x)dx

 E Y    yf Y ( y )dy



Var[ X ]  E[( X   X ) ]   ( x   X ) f X ( x)dx
2
2

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Y
Y=g(X)

E[ g ( X )]   g ( x) f X ( x)dx
y

 E Y    yf Y ( y )dy


x1
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x2
x3
X
Dept of Bioenvironmental Systems Engineering
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Theorem
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Chebyshev Inequality
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
The Chebyshev inequality gives a bound,
which does not depend on the distribution
of X, for the probability of particular
events described in terms of a random
variable and its mean and variance.
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Probability density functions
of discrete random variables







Discrete uniform distribution
Bernoulli distribution
Binomial distribution
Negative binomial distribution
Geometric distribution
Hypergeometric distribution
Poisson distribution
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Discrete uniform distribution
 1
1
x  1,2,, N
f X ( x; N )   N
 I 1, 2,, N  ( x)
N
 0 otherwise
N ranges over the possible integers.
E[ X ]  ( N  1) / 2
Var[ X ]  ( N  1) / 12
2
N
1
mX (t )   e
N
j 1
jt
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Bernoulli distribution
 p x (1  p)1 x
f X ( x; p)  
0
x  0 or 1
otherwise
 p x (1  p)1 x I0,1( x)
0  p 1
1-p is often denoted by q.
E[ X ]  p Var[ X ]  pq
mX (t )  pe  q
t
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Binomial distribution

Binomial distribution represents the
probability of having exactly x success in n
independent and identical Bernoulli trials.
 n  x
  p (1  p) n  x
f X ( x; n, p)   x 
 0

E[ X ]  np
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x  0,1,, n
otherwise
n x
   p (1  p) n  x I0,1,, n ( x)
 x
Var[ X ]  np(1  p)  npq
mX (t )  (q  pet ) n
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Negative binomial distribution


Negative binomial distribution represents the
probability of having exactly r success in x
independent and identical Bernoulli trials.
Unlike the binomial distribution for which the
number of trials is fixed, the number of successes is
fixed and the number of trials varies from
experiment to experiment. The negative binomial
random variable represents the number of trials
needed to obtain exactly r successes.
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 x  1
(1  p) x  r p r
f X ( x; r , p)  
 r  1
r  1,2,; x  r , r  1,
E[ X ]  r / p
Var[ X ]  rq / p
2
mX (t )  ( pe ) /(1  qe )
t r
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t r
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Geometric distribution

Geometric distribution represents the
probability of obtaining the first success in x
independent and identical Bernoulli trials.
f X ( x; p)  (1  p)
E[ X ]  1 / p
x 1
x  1,2,3,
p
Var[ X ]  q / p
2
mX (t )  ( pe ) /(1  qe )
t
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t
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Hypergeometric distribution
  K  M  K 

  
  x  n  x  for x  0,1,, n
f X ( x; M , K , n)  
M 
 

n


0
otherwise
where M is a positive integer, K is a
nonnegative integer that is at most M, and n is
a positive integer that is at most M.
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
Let X denote the number of defective products
in a sample of size n when sampling without
replacement from a box containing M products,
K of which are defective.
E[ X ]  nK / M
K M K M n
Var[ X ]  n 

M
M
M 1
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Poisson distribution

The Poisson distribution provides a realistic model
for many random phenomena for which the number
of occurrences within a given scope (time, length,
area, volume) is of interest. For example, the number
of fatal traffic accidents per day in Taipei, the
number of meteorites that collide with a satellite
during a single orbit, the number of defects per unit
of some material, the number of flaws per unit
length of some wire, etc.
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
e 
f X ( x;  ) 
x  0,1,2,
x!

x
e 

I0,1, ( x)   0
x!
E[ X ]   Var[ X ]  
x
 ( e t 1)
mX (t )  e
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Assume that we are observing the
occurrence of certain happening in time,
space, region or length. Also assume that
there exists a positive quantity  which
satisfies the following properties:
1.
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2.
3.
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Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University
Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University
Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University
Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University
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P0 (t )  e
t
1
0.8
0.6
0.4
0.2
0
0
5
alpha=0.05
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10
15
20
alpha=0.1
25
30
35
alpha=0.2
40
45
50
alpha=0.5
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Comparison of Poisson and
Binomial distributions
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
Example
Suppose that the average number of telephone calls
arriving at the switchboard of a company is 30 calls
per hour.
(1) What is the probability that no calls will arrive
in a 3-minute period?
(2) What is the probability that more than five calls
will arrive in a 5-minute interval?
Assume that the number of calls arriving during
any time period has a Poisson distribution.
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Assuming time is
measured in minutes.
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Assuming time is
measured in seconds.
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



The first property provides the basis for
transferring the mean rate of occurrence
between different observation scales.
The “small time interval of length h” can be
measured in different observation scales.
h  h represents the time length measured
i
in scale of  i .
 i is the mean rate of occurrence when
observation scale  i is used.
i
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
If the first property holds for various
observation scales, say h ,, hn , then it
implies the probability of exactly one
happening in a small time interval h can be
approximated by
1h   2 h     n h
1
1
2
n
 1  h    2  h      n  h 
 1 
 2 
 n 
p

The probability of more than one
happenings in time interval h is negligible.
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
probability that more than five calls will
arrive in a 5-minute interval
=1 - P0 (5)  P1 (5)  P2 (5)  P3 (5)  P4 (5)  P5 (5)
=0.042021
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Probability density functions of
continuous random variables






Uniform or rectangular distribution
Normal distribution (also known as the
Gaussian distribution)
Exponential distribution (or negative
exponential distribution)
Gamma distribution (Pearson Type III)
Chi-squared distribution
Lognormal distribution
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Uniform or rectangular
distribution
1
f X ( x; a, b) 
I [ a ,b ] ( x )
(b  a)
E[ X ]  (a  b) / 2
Var[ X ]  (b  a) 2 / 12
e bt  e at
m X (t ) 
(b  a)t
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PDF of U(a,b)
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Normal distribution
(Gaussian distribution)
f X ( x;  , ) 
E[ X ]  
Var[ X ]  
m X (t )  e
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1
e
2 
1   x   2
 
2   2



2
t  2t 2 / 2
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Commonly used values of
normal distribution
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Exponential distribution (negative
exponential distribution)
f X ( x;  )  ex I[ 0,) ( x) ,  0.
Mean rate of occurrence in a Poisson
E[ X ]  1 / 
process.
Var[ X ]  1 / 2
mX (t ) 
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
 t
for t  
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Gamma distribution
1
 1  x / 
f X ( x; ,  ) 
x
e
I[ 0, ) ( x) ,  0,   0.

( ) 
E[ X ]  
1 /  represents the mean rate of occurrence in a
Poisson process. 1 /  is equivalent to  in the
exponential density.
2
Var[ X ]  
mX (t )  (1  t )
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Hydrology and Spatial Modeling

for t  1/  .
Dept of Bioenvironmental Systems Engineering
National Taiwan University


The exponential distribution is a
special case of gamma distribution
with   1.
The sum of n independent
identically distributed exponential
random variables with parameter 
has a gamma distribution with
parameters   n and   1 /  .
Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University
Pearson Type III
distribution
1  x  
f X ( x) 


 (  )   



2
   
 
 1
e
 x  


  
,   x  
2
    
 ,  and  are mean, standard deviation and
skewness coefficient of X, respectively.
It reduces to Gamma distribution if  = 0.
Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University


The Pearson Type III distribution is
widely applied in stochastic hydrology.
Total rainfall depths of storm events can
be characterized by the Pearson Type III
distribution.
Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University
Chi-squared distribution
1
1
f X ( x; k ) 
 
(k / 2)  2 
E[ X ]  k

k /2
x ( k / 2 ) 1e  x / 2 I[ 0, ) ( x) , k  1,2,.
Var[ X ]  2k
m X (t )  (1  2t )  k / 2 for t  1 / 2.
The chi-squared distribution is a
special case of gamma distribution
with   k / 2 and   2.
Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University
Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University
Log-Normal Distribution
Log-Pearson Type III Distribution


A random variable X is said to have a lognormal distribution if Log(X) is
distributed with a normal density.
A random variable X is said to have a
Log-Pearson type III distribution if
Log(X) has a Pearson type III
distribution.
Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University
Lognormal distribution
1
f X ( x;  , ) 
e
x 2 
E[ X ]  e
1  ln x   2 
 

2
2 


I ( 0, ) ( x)
 ( 2 / 2)
Var[ X ]  e
Lab for Remote Sensing
Hydrology and Spatial Modeling
2  2 2
e
2   2
Dept of Bioenvironmental Systems Engineering
National Taiwan University
Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University
Approximations between
random variables



Approximation of binomial
distribution by Poisson distribution
Approximation of binomial
distribution by normal distribution
Approximation of Poisson
distribution by normal distribution
Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University
Approximation of binomial
distribution by Poisson distribution
Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University
Approximation of binomial
distribution by normal
distribution

Let X have a binomial distribution with
parameters n and p. If n   , then for fixed
a<b,




X  np
P a 
 b  P np  a npq  X  np  b npq   (b)   (a)
npq


(x) is the cumulative distribution function
of the standard normal distribution.
Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University

It is equivalent to say that as n approaches
infinity X can be approximated by a normal
distribution with mean np and variance npq.
Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University
Approximation of Poisson
distribution by normal
distribution

Let X have a Poisson distribution with
parameter . If   , then for fixed a<b


X 


P a 
 b  P   a   X    b   (b)  (a)



Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University

It is equivalent to say that as  approaches
infinity X can be approximated by a normal
distribution with mean  and variance .
Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University
Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University
Example

Suppose that two fair dice are tossed
600 times. Let X denote the number
of times of a total of 7 occurs. What
is the probability that P[90  X  110] ?
Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University
Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University
Transformation of random
variables

[Theorem] Let X be a continuous RV
with density fx. Let Y=g(X), where g is
strictly monotonic and differentiable. The
density for Y, denoted by fY, is given by
1
dg ( y)
f Y ( y)  f X ( g ( y))
dy
1
Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University

Proof: Assume that Y=g(X) is a strictly
monotonic increasing function of X.
Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University
Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University
Using the moment ratio diagram
(MRD) for goodness-of-fit (GOF) test



A two dimensional plot of coefficient of
skewness ( 1 ) vs coefficient of kurtosis( 2 ) is
called a moment ratio diagram.
An MRD uniquely defines the distribution
types of individual random variables.
By examining scattering of sample moment
ratios ( b1 , b2 ) we can identify the distribution
type for the random samples.
Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University
Product (ordinary)
moment ratio diagram
Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University
Simulation


Given a random variable X with CDF FX(x),
there are situations that we want to obtain a set
of n random numbers (i.e., a random sample of
size n) from FX(.) .
The advances in computer technology have made
it possible to generate such random numbers
using computers. The work of this nature is
termed “simulation”, or more precisely
“stochastic simulation”.
Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University
Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University
Pseudo-random number
generation

Pseudorandom number generation
(PRNG) is the technique of
generating a sequence of numbers
that appears to be a random
sample of random variables
uniformly distributed over (0,1).
Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University

A commonly applied approach of PRNG
starts with an initial seed and the
following recursive algorithm (Ross,
2002)
xn  axn 1 modulo m
where a and m are given positive
integers, and where the above means
that axn 1 is divided by m and the
remainder is taken as the value of xn .
Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University


The quantity xn / m is then taken as an
approximation to the value of a uniform
(0,1) random variable.
Such algorithm will deterministically
generate a sequence of values and
repeat itself again and again.
Consequently, the constants a and m
should be chosen to satisfy the
following criteria:
Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University



For any initial seed, the resultant
sequence has the “appearance” of
being a sequence of independent
uniform (0,1) random variables.
For any initial seed, the number of
random variables that can be
generated before repetition begins is
large.
The values can be computed
efficiently on a digital computer.
Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University

A guideline for selection of a and m
is that m be chosen to be a large
prime number that can be fitted to
the computer word size. For a 32bit word computer, m = 231  1 and
a = 75 result in desired properties
(Ross, 2002).
Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University
Simulating a continuous
random variable

probability integral transformation
Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University
Lab for Remote Sensing
Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University