ch3

1. Multionariate ch3 Random Variables Multionariate Random Variables

2. 3.2 Probability mass/density functions Probability mass/density functions

3. a) Discrete Multivariate r.v Definition 3.4 The joint pmf Suppose the r.v. X and Y are discrete. is defined by

4. The Joint pmf of X and Y

5. Solution: Example 3.1 A bag contains 3 black, 2 white and 1 red balls. 2 balls are chosen at random without replacement. Let Find the joint probability mass function

6. Solution: The Joint cdf of X and Y Example 3.2 The joint pmf of X and Y is given by Find the cdf of X and Y

7. Definition 3.5 The marginal Suppose the r.v. X and Y are discrete. pmfis defined by The marginal pmfof X and Y

9. Themarginal cdf of X and Y

10. Solution: Example 3.3 (Example 3.1) A bag contains 3 black, 2 white and 1 red balls. 2 balls are chosen at random without replacement. Let Find the joint pmf and marginal pmf of

11. b) Continuous Multivariate r.v. 1) Double integrals How do we calculate the double integral above? ① R is a region bounded by the curves where ② R is a region bounded by the curves where

12. Example 3.4 Let R is the triangle bounded by Define Find the volume

13. The nonnegative function is a joint pdf of 2) Joint probability density functions Definition 3.6 the continuous random variable X and Y if for all The properties of f (x , y):

14. i.e. the joint pdf is the derivative of (2) The relationship between the joint cdf and pdf is with respect to x and y. (3) We can calculate the probability that (X,Y) falls in a rectangle as below

15. (3) We can calculate the probability that (X,Y) falls in a rectangle as below More generally

16. The marginal pdf of X and Y The marginal cdf of X and Y

17. Example 3.5 Let X and Y denote the proportion of time out of one working day that two employees, A and B, spend performing their assigned tasks. The joint pdf of X and Y is given by (3) Find the marginal pdf of X and Y.

18. 小 结 联合分布 边缘分布

19. A bag contains 3 black, 2 white and 1 red balls. 2 balls are chosen at random without replacement. Let Find the joint probability mass function The joint pmf of X and Y is given by Solution: 0 1 2 0 3/15 3/15 0 1 2 2/15 6/15 0 1/15 0 0

20. Solution:

22. So,the cdf of ( X ,Y )is given by

23. A bag contains 3 black, 2 white and 1 red balls. 2 balls are chosen at random without replacement. Let Find the joint pmf and marginal pmf of Solution: The joint pmf of X and Y is given by 0 1 2 0 3/15 3/15 0 1 2 2/15 6/15 0 1/15 0 0

24. 0 1 2 0 3/15 3/15 0 1 2 2/15 6/15 0 1/15 0 0 The marginal distributions of X and Y are

25. The joint pmf and marginal pmf of X and Y is given by 0 1 2 0 1 2 0 3/15 3/15 + + + + + 2/15 6/15 0 + + + + + 1/15 0 0 + + Joint Marginal