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STATISTICS Joint and Conditional Distributions Professor Ke-Sheng Cheng Department 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. 1/31/2014 2 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Discrete joint density 1/31/2014 3 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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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. 1/31/2014 5 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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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. 1/31/2014 6 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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1/31/2014 7 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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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. 1/31/2014 8 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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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 1/31/2014 9 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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1/31/2014 10 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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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 1/31/2014 11 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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1/31/2014 12 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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1/31/2014 15 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Stochastic independence of random variables 1/31/2014 16 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Expectation of function of a k-dimensional discrete random variable 1/31/2014 17 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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1/31/2014 18 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Covariance 1/31/2014 19 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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1/31/2014 20 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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If two random variables X and Y are independent, then Therefore, 1/31/2014 21 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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However, does not imply that two random variables X and Y are independent. 1/31/2014 22 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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A measure of linear correlation: Pearson coefficient of correlation 1/31/2014 23 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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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. 1/31/2014 24 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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1/31/2014 25 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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1/31/2014 26 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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1/31/2014 27 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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1/31/2014 28 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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1/31/2014 30 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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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) 1/31/2014 31 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Bivariate Normal Distribution Bivariate normal density function 1/31/2014 32 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Conditional normal density 1/31/2014 33 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Bivariate normal simulation I. Using the conditional density 1/31/2014 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ. 35

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(x,y) scatter plot 1/31/2014 38 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Histogram of X 1/31/2014 39 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Histogram of Y 1/31/2014 40 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Bivariate normal simulation II. Using the PC Transformation 1/31/2014 41 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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1/31/2014 42 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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1/31/2014 43 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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1/31/2014 44 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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(x,y) scatter plot 1/31/2014 45 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Histogram of X 1/31/2014 46 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Histogram of Y 1/31/2014 47 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Multivariate normal simulation using R The mvtnorm package in R dmvnorm rmvnorm pmvnorm qmvnorm 1/31/2014 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ. 48

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Conceptual illustration of Bivariate gamma simulation 1/31/2014 49 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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