1. STATISTICS Joint and Conditional Distributions Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University

2. 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. 12/20/2015 2 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

3. Discrete joint density 12/20/2015 3 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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5. Marginal discrete density If X and Y are bivariate joint discrete random variables, then and are called marginal discrete density functions. 12/20/2015 5 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

6. 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. 12/20/2015 6 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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8. Marginal continuous probability density function If X and Y are bivariate joint continuous random variables, then and are called marginal probability density functions. 12/20/2015 8 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

9. 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 12/20/2015 9 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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11. 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 12/20/2015 11 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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15. Stochastic independence of random variables 12/20/2015 15 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

16. Expectation of function of a k-dimensional discrete random variable 12/20/2015 16 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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18. Conditional Expectation 12/20/2015 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ. 18

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20. Expectation of a Random Sum of Random Variables Let N be a random variable which can assume positive integer values 1, 2, 3.... Let X i be a sequence of independent random variables which are also independent of N and have a common mean E[X] independent of i. Then the expectation of the sum of N X i ’s can be expressed as 12/20/2015 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ. 20 Homework problem

21. Covariance 12/20/2015 21 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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23. If two random variables X and Y are independent, then Therefore, 12/20/2015 23 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

24. However, does not imply that two random variables X and Y are independent. 12/20/2015 24 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

25. A measure of linear correlation: Pearson coefficient of correlation 12/20/2015 25 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

26. 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. 12/20/2015 26 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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33. 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) 12/20/2015 33 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

34. Bivariate Normal Distribution Bivariate normal density function 12/20/2015 34 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

35. Conditional normal density 12/20/2015 35 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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38. Bivariate normal simulation I. Using the conditional density 12/20/2015 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ. 38

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41. (x,y) scatter plot 12/20/2015 41 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

42. Histogram of X 12/20/2015 42 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

43. Histogram of Y 12/20/2015 43 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

44. Bivariate normal simulation II. Using the PC Transformation 12/20/2015 44 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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48. (x,y) scatter plot 12/20/2015 48 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

49. Histogram of X 12/20/2015 49 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

50. Histogram of Y 12/20/2015 50 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

51. Multivariate normal simulation using R The mvtnorm package in R dmvnorm rmvnorm pmvnorm qmvnorm 12/20/2015 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ. 51

52. Conceptual illustration of Bivariate gamma simulation 12/20/2015 52 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.