# STATISTICS Joint and Conditional Distributions

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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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Bivariate normal simulation I. Using the conditional density

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

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