环境水文学Chapter9.ppt
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• Discrete variables
– Random variables that are restricted to specific, incremental values
June 11, 2009
Principle of Hydrology, BNU
4
Probability Distribution
hydrology • Probability distribution applied to
hydrology • Frequency analysis • Pearson Type III method
June 11, 2009
Principle of Hydrology, BNU
2
Random Variables
11
Lognormal Distribution
f
(y)
1
x y
2
exp
(
y y 2 y 2
)2
y ln(x)
June 11, 2009
Principle of Hydrology, BNU
12
Pearson III Distribution
f
(x)
( )
(x
a ) e 1 ( xa0 ) 0
• Definition
– A graph of probability in which random variables, either continuous or discrete, are characterized by the distribution of probabilities attached to the specific values that the variable may take on.
June 11, 2009
Principle of Hydrology, BNU
9
Continuous Probability Distribution
• Normal distribution • Lognormal distribution • Gamma distribution
June 11, 2009
F(x) PX x
June 11, 2009
Principle of Hydrology, BNU
6
Statistics
• Mean • Variance/standard deviation • Coefficient of variation • Skewness/Cofficient of skewness
Chapter 9 Statistical Methods
and Frequency Analysis
Prof. Zongxue XU
June 11, 2009
Principle of Hydrology, BNU
1
Objectives
• Probability, frequency, return period • Probability theory applied to
s x
June 11, 2009
Principle of Hydrology, BNU
8
Coefficient of Skewness
Cs
n (n 1)(n 2)s3
n
( xi
i 1
x)3
• For symmetrical distribution: Cs=0 • For right skewness: Cs>0 • For left skewness: Cs<0
Principle of Hydrology, BNU
10
Normal Distribution
• Normal distribution
f (x)
1wk.baidu.com
e
(
x )2 2 2
2
• Standard normal distribution
f (z)
1
z2
e2
2
June 11, 2009
Principle of Hydrology, BNU
June 11, 2009
Principle of Hydrology, BNU
7
Mean/Variance
• Mean
x
1 n
n i 1
xi
• Variance
•
Standard
s2 1
n 1
deviation
n i 1
( xi
x)2
s
• Coefficient of variation
cV
• Pearson III distribution has been widely used in the flood design in China
June 11, 2009
Principle of Hydrology, BNU
13
Frequency Analysis
• Definition
– The frequency of a hydrologic event is the probability that some value of a discrete variable will occur or some value of continuous variable will be equaled or exceeded in any given year
• Definition
– A variable that demonstrates variability that is not sufficiently explained by analytical measures of physical processes
• Examples
– Precipitation
PX x1 or f (x)
June 11, 2009
Principle of Hydrology, BNU
5
Cumulative Distribution Function
• Definition (CDF)
– A graph of the probability that any outcome in X is less than or equal to a stated, limiting value x.
– Runoff
–…
June 11, 2009
Principle of Hydrology, BNU
3
Continuous/Discrete Random Variables
• Continuous variables
– Random variables that may take on all values in the range of occurrence, including figures differing only by an infinitesimal amount
– Random variables that are restricted to specific, incremental values
June 11, 2009
Principle of Hydrology, BNU
4
Probability Distribution
hydrology • Probability distribution applied to
hydrology • Frequency analysis • Pearson Type III method
June 11, 2009
Principle of Hydrology, BNU
2
Random Variables
11
Lognormal Distribution
f
(y)
1
x y
2
exp
(
y y 2 y 2
)2
y ln(x)
June 11, 2009
Principle of Hydrology, BNU
12
Pearson III Distribution
f
(x)
( )
(x
a ) e 1 ( xa0 ) 0
• Definition
– A graph of probability in which random variables, either continuous or discrete, are characterized by the distribution of probabilities attached to the specific values that the variable may take on.
June 11, 2009
Principle of Hydrology, BNU
9
Continuous Probability Distribution
• Normal distribution • Lognormal distribution • Gamma distribution
June 11, 2009
F(x) PX x
June 11, 2009
Principle of Hydrology, BNU
6
Statistics
• Mean • Variance/standard deviation • Coefficient of variation • Skewness/Cofficient of skewness
Chapter 9 Statistical Methods
and Frequency Analysis
Prof. Zongxue XU
June 11, 2009
Principle of Hydrology, BNU
1
Objectives
• Probability, frequency, return period • Probability theory applied to
s x
June 11, 2009
Principle of Hydrology, BNU
8
Coefficient of Skewness
Cs
n (n 1)(n 2)s3
n
( xi
i 1
x)3
• For symmetrical distribution: Cs=0 • For right skewness: Cs>0 • For left skewness: Cs<0
Principle of Hydrology, BNU
10
Normal Distribution
• Normal distribution
f (x)
1wk.baidu.com
e
(
x )2 2 2
2
• Standard normal distribution
f (z)
1
z2
e2
2
June 11, 2009
Principle of Hydrology, BNU
June 11, 2009
Principle of Hydrology, BNU
7
Mean/Variance
• Mean
x
1 n
n i 1
xi
• Variance
•
Standard
s2 1
n 1
deviation
n i 1
( xi
x)2
s
• Coefficient of variation
cV
• Pearson III distribution has been widely used in the flood design in China
June 11, 2009
Principle of Hydrology, BNU
13
Frequency Analysis
• Definition
– The frequency of a hydrologic event is the probability that some value of a discrete variable will occur or some value of continuous variable will be equaled or exceeded in any given year
• Definition
– A variable that demonstrates variability that is not sufficiently explained by analytical measures of physical processes
• Examples
– Precipitation
PX x1 or f (x)
June 11, 2009
Principle of Hydrology, BNU
5
Cumulative Distribution Function
• Definition (CDF)
– A graph of the probability that any outcome in X is less than or equal to a stated, limiting value x.
– Runoff
–…
June 11, 2009
Principle of Hydrology, BNU
3
Continuous/Discrete Random Variables
• Continuous variables
– Random variables that may take on all values in the range of occurrence, including figures differing only by an infinitesimal amount