June 11, 2009
Principle of Hydrology, BNU
7
Mean/Variance• Meanx1 nn i 1
xi
• Variance
•
Standard
s2 1
n 1
deviation
n i 1
( xi
x)2
s
• Coefficient of variation
cV
Principle of Hydrology, BNU
10
Normal Distribution
• Normal distribution
f (x)
1
e
(
x )2 2 2
2
• Standard normal distribution
f (z)
1
z2
e2
2
June 11, 2009
Principle of Hydrology, BNU
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.
• Discrete variables
– Random variables that are restricted to specific, incremental values
June 11, 2009
Principle of Hydrology, BNU
4
Probability Distribution
June 11, 2009
Principle of Hydrology, BNU
9
Continuous Probability Distribution
• Normal distribution • Lognormal distribution • Gamma distribution
June 11, 2009
• 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.
– 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
F(x) PX x
June 11, 2009
Principle of Hydrology, BNU
6
Statistics
• Mean • Variance/standard deviation • Coefficient of variation • Skewness/Cofficient of skewness
hydrology • Probability distribution applied to
hydrology • Frequency analysis • Pearson Type III method
June 11, 2009
Principle of Hydrology, BNU
2
Random Variables
• 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
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
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
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