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![Page 1: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/1.jpg)
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
Lab for Remote Sensing Hydrology and Spatial
Modeling
STATISTICS Joint and Conditional
Distributions
Professor Ke-Sheng ChengDepartment of Bioenvironmental Systems
EngineeringNational Taiwan University
![Page 2: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/2.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
Joint cumulative distribution function
Let be k random variables all defined on the same probability space ( ,A, P[]). The joint cumulative distribution function of , denoted by , is defined as
for all .
kXXX ,,, 21
kXXX ,,, 21 ),,(,,1 kXXF
],,[ 2211 kk xXxXxXP ),,( 21 kxxx
![Page 3: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/3.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
Discrete joint density
![Page 4: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/4.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
![Page 5: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/5.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
![Page 6: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/6.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
Marginal discrete density
If X and Y are bivariate joint discrete random variables, then and are called marginal discrete density functions.
)(Xf )(Yf
}:{, ),()(
ki xxiiiYXkX yxfxf
}:{, ),()(
ki yyiiiYXkY yxfyf
0),( yxf XY x y
XY yxf 1),(
![Page 7: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/7.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
Continuous Joint Density Function
The k-dimensional random variable ( ) is defined to be a k-dimensional continuous random variable if and only if there exists a function such that
for all . is defined to be the joint
probability density function.
kXXX ,, 21
0),,(,,1 kXXf
k
x x
kXXkXX duduuufxxFk
kk 11,,1,,
1
11),,(),,(
),,( 21 kxxx
),,(,,1 kXXf
![Page 8: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/8.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
0),,( 1,,1kXX xxf
k
1),,( 11,,1
kkXX dxdxxxfk
],,,[ 222111 kkk bXabXabXaP
k
b
a
b
a kXX dxdxxxfk
kk
11,,
1
11
),,(
![Page 9: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/9.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
Marginal continuous probability density function
If X and Y are bivariate joint continuous random variables, then and are called marginal probability density functions.
)(Xf )(Yf
dyyxfxf XYX ),()(
dxyxfyf XYY ),()(
![Page 10: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/10.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
Conditional distribution functions for discrete random variables
If X and Y are bivariate joint discrete random variables with joint discrete density function
, then the conditional discrete density function of Y given X=x, denoted by
or , is defined to be
),( XYf
)|(| xf XY )(| xXYf
)(
),()|(| xf
yxfxyf
X
XYXY
![Page 11: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/11.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
}:{|| )|(]|[)|(
yyjjXYXY
j
xyfxXyYPxyF
![Page 12: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/12.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
Conditional distribution functions for continuous random variables
If X and Y are bivariate joint continuous random variables with joint continuous density function , then the conditional probability density function of Y given X=x, denoted by or , is defined to be
),( XYf
)|(| xf XY )(| xXYf
)(
),()|(| xf
yxfxyf
X
XYXY
![Page 13: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/13.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
![Page 14: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/14.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
dyyxfxf
dyxf
yxfdyxyf
XYX
X
XYXY
),()(
1
)(
),()|(|
1)(
)(
xf
xf
X
X
![Page 15: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/15.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
![Page 16: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/16.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
![Page 17: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/17.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
![Page 18: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/18.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
Stochastic independence of random variables
![Page 19: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/19.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
Expectation of function of a k-dimensional discrete random variable
![Page 20: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/20.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
![Page 21: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/21.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
![Page 22: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/22.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
Covariance
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Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
YXXYEYXCov ][),(
![Page 24: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/24.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
If two random variables X and Y are independent, then .0),( YXCov
YXXYEYXCov ][),(
YXYX
YX
XY
dyyyfdxxxf
dxdyyfxxyf
dxdyyxxyfXYE
)()(
)()(
),()(
Therefore, .0),( YXCov
![Page 25: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/25.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
However, does not imply that two random variables X and Y are independent.
0),( YXCov
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Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
A measure of linear correlation:Pearson coefficient of correlation
YXXY
YXCovYXCorrel
),(
),(
11 XY
![Page 27: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/27.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
Covariance and Correlation Coefficient
Suppose we have observed the following data. We wish to measure both the direction and the strength of the relationship between Y and X.
![Page 28: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/28.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
![Page 29: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/29.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
![Page 30: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/30.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
![Page 31: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/31.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
![Page 32: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/32.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
![Page 33: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/33.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
![Page 34: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/34.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
![Page 35: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/35.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
Examples of joint distributions
Duration and total depth of storm events. (bivariate gamma, non-causal relation)
Hours spent for study and test score. (causal relation)
![Page 36: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/36.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
Bivariate Normal Distribution
Bivariate normal density function
1
2
1
21
2
1)(),(
zz
ZXY
T
ezfyxf
![Page 37: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/37.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
Conditional normal density
2
22|1
)()(
2
1exp
)1(2
1)|(
Y
XX
YY
Y
XY
xy
xyYf
)(| yf xXY
![Page 38: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/38.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
![Page 39: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/39.jpg)
Bivariate normal simulation I. Using the conditional density
2
22|1
)()(
2
1exp
)1(2
1)|(
Y
XX
YY
Y
XY
xy
xyYf
![Page 40: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/40.jpg)
![Page 41: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/41.jpg)
(x,y) scatter plot
![Page 42: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/42.jpg)
Histogram of X
![Page 43: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/43.jpg)
Histogram of Y
![Page 44: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/44.jpg)
Bivariate normal simulation II. Using the PC Transformation
![Page 45: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/45.jpg)
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![Page 48: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/48.jpg)
(x,y) scatter plot
![Page 49: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/49.jpg)
Histogram of X
![Page 50: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/50.jpg)
Histogram of Y
![Page 51: Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.](https://reader033.fdocuments.net/reader033/viewer/2022061306/55147f56550346d36e8b48c5/html5/thumbnails/51.jpg)
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems EngineeringNational Taiwan University
Conceptual illustration of Bivariate gamma simulation