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STATISTICS
Sampling and Sampling
Distributions
Professor Ke-Sheng Cheng
Department of Bioenvironmental Systems Engineering
National Taiwan University
Random sample
• Let the random variables X1, X2, …, Xn have a
joint density f X1 , X 2 ,, X n (,, ,) that factors as
follows:
f X1 , X 2 ,, X n ( x1 , x 2 , x n )  f ( x1 ) f ( x 2 ) f ( x n )
where f () is the common density of each Xi .
Then (X1, X2, …, Xn) is defined to be a random
sample of size n from a population with
density f () .
5/6/2017
Laboratory for Remote Sensing Hydrology and Spatial Modeling,
Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
2
• If X1, X2, …, Xn is a random sample of size n
from f () , then X1, X2, …, Xn are stochastically
independent.
• Histogram -- A frequency (or relative frequency)
plot of observed data is called a frequency
histogram (or relative frequency histogram).
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Laboratory for Remote Sensing Hydrology and Spatial Modeling,
Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Frequency Histogram
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
4
Cumulative frequency
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Relative cumulative frequency
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Laboratory for Remote Sensing Hydrology and Spatial Modeling,
Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Statistic
• A statistic is a function of observable random
variables, which is itself an observable random
variable and does not contain any unknown
parameters.
• A statistic must be observable because we
intend to use it to make inferences about the
density functions of the random variables.
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Laboratory for Remote Sensing Hydrology and Spatial Modeling,
Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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• For example, if a random variable has a
probability density function N (,  2 ) where  and
 are unknown, then  X   is not a statistic.
• If a statistic is not observable, then it can not be
used to inference the parameters of the density
function.
n
i 1
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2
i
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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• An observation of random sample of size n can
be regarded as n independent observations of a
random variable.
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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• One of the central problems in statistics is to
find suitable statistics to represent parameters
of the probability distribution function of a
random variable.
Sample {x1 ,, xn }
Population N (  , 2 )
Statistics ( x , s 2 )
2
Parameters (  ,  )
Observable
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Unknown
Laboratory for Remote Sensing Hydrology and Spatial Modeling,
Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
10
Sample moments
• Let X1, X2, …, Xn be a random sample from the
density f () . Then the rth sample moment about
0 is defined as
n
1
r
'
Mr   Xi
n i 1
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Laboratory for Remote Sensing Hydrology and Spatial Modeling,
Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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• In particular, if r = 1, we have the sample
mean X n ; that is,
1 n
Xn   Xi
n i 1
• Also, the rth sample moment about the sample
mean is defined as
1 n
r
Mr   (Xi  Xn)
n i 1
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Laboratory for Remote Sensing Hydrology and Spatial Modeling,
Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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• Theorem – Let X1, X2, …, Xn be a random
sample from the density f () . The expected
value of the rth sample moment about 0 is equal
'
'
th
to the r population moment; i.e., E[ M r ]   r
Also,
Var[ M r' ]
1
1 '
2r
r 2
 {E[ X ]  ( E[ X ]) }  [  2 r  (  r' ) 2 ]
n
n
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Laboratory for Remote Sensing Hydrology and Spatial Modeling,
Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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• Special case: r=1
1
2
2
Var[ X ]  {E[ X ]  ( E[ X ]) }
n
1 '
Var
(
X
)
' 2
2
 [  2  ( 1 ) ] 
X /n
n
n
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Laboratory for Remote Sensing Hydrology and Spatial Modeling,
Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
14
Sample statistics
• Let X1, X2, …, Xn be a random sample from the
distribution of a random variable X. Sample
mean and sample variance of the distribution are
respectively defined to be
n
1
X   Xi
n i 1
5/6/2017
n
1
2
2
S 
(Xi  X )

n  1 i 1
Laboratory for Remote Sensing Hydrology and Spatial Modeling,
Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Estimating the mean
• Given a random sample x1 , x2 , xn from a probability
density function f(.) with unknown mean μ and finite
variance σ2, we want to estimate the mean using the
random sample.
• Using only a finite number of values of X (a random
sample of size n), can any reliable inferences be made
about E(X), the average of an infinite number of values
of X?
• Will the estimate be more reliable if the size of the
random sample is larger?
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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R-program demonstration
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Mean of sample means w.r.t. sample
size
60.2
60.15
60.1
60.05
60
59.95
59.9
59.85
59.8
0
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1000
2000
3000
4000
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
5000
23
Mean of sample standard deviations w.r.t.
sample size
20.02
20
19.98
19.96
19.94
19.92
19.9
19.88
19.86
19.84
0
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1000
2000
3000
4000
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
5000
24
Standard deviation of sample means w.r.t. sample size
5
4.5
4
3.5
What is the theoretical basis?
3
y = 19.938x-0.4998
Y=f(x)=?
R = 0.9995
2.5
2
2
1.5
1
0.5
0
0
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1000
2000
3000
4000
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
5000
25
Histograms of sample mean and standard deviation
ns=30
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Histograms of sample mean and standard deviation
ns=5000
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Weak Law of Large Numbers
(WLLN)
• Let f(.) be a density with mean μ and variance
σ2, and let X n be the sample mean of a random
sample of size n from f(.). Let ε and δ be any
two specified numbers satisfying ε>0 and 0<δ<1.
2

If n is any integer greater than
, then
2
 
P[  X n     ]  1  
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Recall the theorem
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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• (Example) Suppose that some distribution
with an unknown mean has its variance equal
to 1. How large a random sample must be
taken such that the probability will be at least
0.95 that the sample mean X n will lie within
0.5 of the population mean?
  1   0.5
2
  1  0.95  0.05
1
n
 80
2
(0.05)(0.5)
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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(Example) How large a random sample
must be taken in order that you are 99%
certain that X n is within 0.5σ of μ?
  0.5
  1  0.992  0.01
n
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
(0.01)(0.5 )
2
 400
Laboratory for Remote Sensing Hydrology and Spatial Modeling,
Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
32
Raingauge network design
• Assuming there are already some raingauge
stations in a catchment, and we are interested in
determining the optimal number of stations that
should exist to achieve a desired accuracy in the
estimation of mean rainfall.
• Two approaches
– (1) Standard deviation of the sample mean should
not exceed a certain portion of the population mean.
– (2) P[     xn     ]  1  
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Criterion 1
Standard deviation of the sample mean should not exceed a
certain portion of the population mean.
X n ~ N (  ,  / n) ,
2
X 
n

n
 CV 
n

  
5/6/2017
  ,
( X n   ) ~ N (0,

2
n
)
 CV
 n

 
2
Laboratory for Remote Sensing Hydrology and Spatial Modeling,
Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Criterion 2
P[     xn     ]  1  
• From the weak law of large numbers,

n 2

2
What assumptions have we made for such approaches of
network design ?
Data independence
What are the practical considerations in monitoring network
design?
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Laboratory for Remote Sensing Hydrology and Spatial Modeling,
Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
35
The Central Limit Theorem
• Let f(.) be a density with mean μ and finite
variance σ2. Let X n be the sample mean of a
random sample of size n from f(.). Then
Zn 
Xn  

n
approaches the standard normal distribution as
n approaches infinity.
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Laboratory for Remote Sensing Hydrology and Spatial Modeling,
Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
36
• The importance of the CLT is the fact that the
mean X n of a random sample from any
distribution with finite variance σ2 and mean μ is
approximately distributed as a normal2 random
variable with mean μ and variance  n .






X n    Zn 
 ~ N  ,

n
n


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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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R-program demonstration
- Central Limit Theorem
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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n=100
n=50
n=25
n=2
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n=10
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Sampling distributions
• Given random samples of certain probability densities,
we often are interested in knowing the probability
densities of sampling statistics.
–
–
–
–
–
–
Poisson distribution
Exponential distribution
Normal distribution
Chi-square distribution
Standard normal and chi-square distributions
Student’s t-distribution
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Poisson distribution
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Exponential distribution
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Normal distribution
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
52
5/6/2017
Laboratory for Remote Sensing Hydrology and Spatial Modeling,
Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
53
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
54
Chi-square distribution
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Standard normal and chi-square
distributions
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Student’s t-distribution
Student’s t distribution with k degrees of freedom
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• The "student's" distribution was published in
1908 by W. S. Gosset. Gosset, however, was
employed at a brewery that forbade the
publication of research by its staff members. To
circumvent this restriction, Gosset used the
name "Student", and consequently the
distribution was named "Student t-distribution.
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Order statistics
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