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

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

Lab for Remote Sensing Hydrology and Spatial Modeling

Dept of Bioenvironmental Systems EngineeringNational Taiwan University

Discrete joint density

Lab for Remote Sensing Hydrology and Spatial Modeling

Dept of Bioenvironmental Systems EngineeringNational Taiwan University

Lab for Remote Sensing Hydrology and Spatial Modeling

Dept of Bioenvironmental Systems EngineeringNational Taiwan University

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),(

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

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

),,(

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 ),()(

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

Lab for Remote Sensing Hydrology and Spatial Modeling

Dept of Bioenvironmental Systems EngineeringNational Taiwan University

}:{|| )|(]|[)|(

yyjjXYXY

j

xyfxXyYPxyF

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

Lab for Remote Sensing Hydrology and Spatial Modeling

Dept of Bioenvironmental Systems EngineeringNational Taiwan University

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

Lab for Remote Sensing Hydrology and Spatial Modeling

Dept of Bioenvironmental Systems EngineeringNational Taiwan University

Lab for Remote Sensing Hydrology and Spatial Modeling

Dept of Bioenvironmental Systems EngineeringNational Taiwan University

Lab for Remote Sensing Hydrology and Spatial Modeling

Dept of Bioenvironmental Systems EngineeringNational Taiwan University

Lab for Remote Sensing Hydrology and Spatial Modeling

Dept of Bioenvironmental Systems EngineeringNational Taiwan University

Stochastic independence of random variables

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

Lab for Remote Sensing Hydrology and Spatial Modeling

Dept of Bioenvironmental Systems EngineeringNational Taiwan University

Lab for Remote Sensing Hydrology and Spatial Modeling

Dept of Bioenvironmental Systems EngineeringNational Taiwan University

Lab for Remote Sensing Hydrology and Spatial Modeling

Dept of Bioenvironmental Systems EngineeringNational Taiwan University

Covariance

Lab for Remote Sensing Hydrology and Spatial Modeling

Dept of Bioenvironmental Systems EngineeringNational Taiwan University

YXXYEYXCov ][),(

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

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

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

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.

Lab for Remote Sensing Hydrology and Spatial Modeling

Dept of Bioenvironmental Systems EngineeringNational Taiwan University

Lab for Remote Sensing Hydrology and Spatial Modeling

Dept of Bioenvironmental Systems EngineeringNational Taiwan University

Lab for Remote Sensing Hydrology and Spatial Modeling

Dept of Bioenvironmental Systems EngineeringNational Taiwan University

Lab for Remote Sensing Hydrology and Spatial Modeling

Dept of Bioenvironmental Systems EngineeringNational Taiwan University

Lab for Remote Sensing Hydrology and Spatial Modeling

Dept of Bioenvironmental Systems EngineeringNational Taiwan University

Lab for Remote Sensing Hydrology and Spatial Modeling

Dept of Bioenvironmental Systems EngineeringNational Taiwan University

Lab for Remote Sensing Hydrology and Spatial Modeling

Dept of Bioenvironmental Systems EngineeringNational Taiwan University

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)

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

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

Lab for Remote Sensing Hydrology and Spatial Modeling

Dept of Bioenvironmental Systems EngineeringNational Taiwan University

Bivariate normal simulation I. Using the conditional density

2

22|1

)()(

2

1exp

)1(2

1)|(

Y

XX

YY

Y

XY

xy

xyYf

(x,y) scatter plot

Histogram of X

Histogram of Y

Bivariate normal simulation II. Using the PC Transformation

(x,y) scatter plot

Histogram of X

Histogram of Y

Lab for Remote Sensing Hydrology and Spatial Modeling

Dept of Bioenvironmental Systems EngineeringNational Taiwan University

Conceptual illustration of Bivariate gamma simulation