1. 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

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

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

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

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

6. 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

7. 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

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

9. 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

10. 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

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

12. 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

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

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

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

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

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

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

19. 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

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

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

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

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

24. 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

25. 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

26. 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

27. 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

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

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

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

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

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

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

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

35. 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

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

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

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

39. Bivariate normal simulation I. Using the conditional density

41. (x,y) scatter plot

42. Histogram of X

43. Histogram of Y

44. Bivariate normal simulation II. Using the PC Transformation

48. (x,y) scatter plot

49. Histogram of X

50. Histogram of Y

51. Conceptual illustration of Bivariate gamma simulation
Lab for Remote Sensing Hydrology and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University