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**STATISTICS Joint and Conditional Distributions**

Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

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

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**Discrete joint density**

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

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**Lab for Remote Sensing Hydrology and Spatial Modeling **

Dept of Bioenvironmental Systems Engineering National Taiwan University

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**Lab for Remote Sensing Hydrology and Spatial Modeling **

Dept of Bioenvironmental Systems Engineering National Taiwan University

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**Marginal discrete density**

If X and Y are bivariate joint discrete random variables, then and are called marginal discrete density functions. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

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**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. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

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**Lab for Remote Sensing Hydrology and Spatial Modeling **

Dept of Bioenvironmental Systems Engineering National Taiwan University

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**Marginal continuous probability density function**

If X and Y are bivariate joint continuous random variables, then and are called marginal probability density functions. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

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

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**Lab for Remote Sensing Hydrology and Spatial Modeling **

Dept of Bioenvironmental Systems Engineering National Taiwan University

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

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**Lab for Remote Sensing Hydrology and Spatial Modeling **

Dept of Bioenvironmental Systems Engineering National Taiwan University

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**Lab for Remote Sensing Hydrology and Spatial Modeling **

Dept of Bioenvironmental Systems Engineering National Taiwan University

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**Lab for Remote Sensing Hydrology and Spatial Modeling **

Dept of Bioenvironmental Systems Engineering National Taiwan University

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**Lab for Remote Sensing Hydrology and Spatial Modeling **

Dept of Bioenvironmental Systems Engineering National Taiwan University

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**Lab for Remote Sensing Hydrology and Spatial Modeling **

Dept of Bioenvironmental Systems Engineering National Taiwan University

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**Stochastic independence of random variables**

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

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**Expectation of function of a k-dimensional discrete random variable**

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

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**Lab for Remote Sensing Hydrology and Spatial Modeling **

Dept of Bioenvironmental Systems Engineering National Taiwan University

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**Lab for Remote Sensing Hydrology and Spatial Modeling **

Dept of Bioenvironmental Systems Engineering National Taiwan University

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**Covariance Lab for Remote Sensing Hydrology and Spatial Modeling**

Dept of Bioenvironmental Systems Engineering National Taiwan University

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**Lab for Remote Sensing Hydrology and Spatial Modeling **

Dept of Bioenvironmental Systems Engineering National Taiwan University

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**If two random variables X and Y are independent, then**

Therefore, Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

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

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

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**A measure of linear correlation: Pearson coefficient of correlation**

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

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**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 Engineering National Taiwan University

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**Lab for Remote Sensing Hydrology and Spatial Modeling **

Dept of Bioenvironmental Systems Engineering National Taiwan University

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**Lab for Remote Sensing Hydrology and Spatial Modeling **

Dept of Bioenvironmental Systems Engineering National Taiwan University

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**Lab for Remote Sensing Hydrology and Spatial Modeling **

Dept of Bioenvironmental Systems Engineering National Taiwan University

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**Lab for Remote Sensing Hydrology and Spatial Modeling **

Dept of Bioenvironmental Systems Engineering National Taiwan University

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**Lab for Remote Sensing Hydrology and Spatial Modeling **

Dept of Bioenvironmental Systems Engineering National Taiwan University

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**Lab for Remote Sensing Hydrology and Spatial Modeling **

Dept of Bioenvironmental Systems Engineering National Taiwan University

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**Lab for Remote Sensing Hydrology and Spatial Modeling **

Dept of Bioenvironmental Systems Engineering National Taiwan University

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**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 Engineering National Taiwan University

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**Bivariate Normal Distribution**

Bivariate normal density function Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

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**Conditional normal density**

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

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**Lab for Remote Sensing Hydrology and Spatial Modeling **

Dept of Bioenvironmental Systems Engineering National Taiwan University

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**Bivariate normal simulation I. Using the conditional density**

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(x,y) scatter plot

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Histogram of X

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Histogram of Y

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**Bivariate normal simulation II. Using the PC Transformation**

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(x,y) scatter plot

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Histogram of X

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Histogram of Y

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**Conceptual illustration of Bivariate gamma simulation**

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

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