Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional

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


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



Discrete joint density







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),(



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



0),,( 1,,1kXX xxf

k

1),,( 11,,1

kkXX dxdxxxfk

],,,[ 222111 kkk bXabXabXaP

k

b

a

b

a kXX dxdxxxfk

kk

11,,

1

11

),,(



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 ),()(



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



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

yyjjXYXY

j

xyfxXyYPxyF



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





dyyxfxf

dyxf

yxfdyxyf

XYX

X

XYXY

),()(

1

)(

),()|(|

1)(

)(

xf

xf

X

X









Stochastic independence of random variables



Expectation of function of a k-dimensional discrete random variable







Covariance



YXXYEYXCov ][),(



If two random variables X and Y are independent, then .0),( YXCov

YXXYEYXCov ][),(

YXYX

YX

XY

dyyyfdxxxf

dxdyyfxxyf

dxdyyxxyfXYE

)()(

)()(

),()(

Therefore, .0),( YXCov



However, does not imply that two random variables X and Y are independent.

0),( YXCov



A measure of linear correlation:Pearson coefficient of correlation

YXXY

YXCovYXCorrel

),(

),(

11 XY



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.

















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)



Bivariate Normal Distribution

Bivariate normal density function

1

2

1

21

2

1)(),(

zz

ZXY

T

ezfyxf



Conditional normal density

2

22|1

)()(

2

1exp

)1(2

1)|(

Y

XX

YY

Y

XY

xy

xyYf

)(| yf xXY



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



Conceptual illustration of Bivariate gamma simulation

Documents

Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional