1. School of Mathematical Science and Computing Technology in CSU
Probability Theory
School of Mathematical Science and
Computing Technology in CSU
Course groups of Probability and Statistics

2. §2.2 Multi-dimensional random variables and their distributions

3. So far, we have only discussed the one-dimensional random variables and their distribution, But using a random variable to describe some random phenomenon is not enough ，so we need to use some random variables to describe it
At target practice, the hit points are determined by a pair of random variables (two coordinates).
The center gravity position of the plane in the air are determined by three random variables (three coordinates) and so on.

4. Two-dimensional discrete random variable
Definition:
If (X, Y) only take a real value of limited or
numbered pairs
So we call it two-dimensional discrete random variable.

5. i, j =1,2, … k=1,2, … k=1,2, … Two-dimensional random variables (X, Y)
One-dimensional random variable X
Discrete
Discrete
the joint probability distribution of
X and Y
the probability distribution of X
i, j =1,2, …
k=1,2, …
k=1,2, …

6. The table form of the joint probability distribution of (X,Y) is as follows:…
x i
y y2 … y j …
p p12 … p1j …
p p22 … p2j …
… … … … …
pi pi2 … p i j …
… … … … … Y
因“五.一”放假,超级链接用于复习基本知识点,以方便后继课

7. The joint distribution of two-dimensional has a comprehensive reflection of the value of two-dimensional random variables (X, Y) and its probability law, while a single random variable X, Y also has its own probability distribution.
The probability distribution of X and Y are known as the edge (probability) distribution of（X,Y）on X or Y.
Then ask: what is the Relation between these two?
Could they be sure with each other？
First look at how to determine the two edge distribution
by the joint distribution

8. So the marginal probability distribution of (X,Y) about X is
In general ，for two-dimensional discrete random variable (X, Y) ，
the joint probability distribution of X and Y is
So the marginal probability distribution of
(X,Y) about X is

9. The same
(j=1,2,...)
in a general way, set:

10. We often write the marginal probability function on the edge of the Joint probability function table, so we get the term of marginal distribution.

11. Example 1 There are two white balls and three
black balls in the bag，to reap the ball twice
question：try to get the joint distribution and marginal distribution of (X,Y), please discuss it at the Situation of back-extraction and no-back.

12. When with back，X and Y is independent with each other；
Solutions：with back
no-back 1 1
When with back，X and Y is independent with each other；
When with no-back, it’ not！

13. The relationship between joint distribution and marginal distribution ：
The marginal distribution can be determined by joint distribution;
While the joint distribution can not be determined by marginal distribution.

14. If Then case A and B is independent.
The definition of that case A and B is independent is： If
Then case A and B is independent.
The independence of the case will be extended to the random variables
The independence of random variables in probability theory is an important concept

15. So call it is independence between X and Y.
If (X,Y) is discrete rand variables，the definition of the independence above equals to：
For all the possible value (xi, yj) of (X,Y), there are:
viz.
So call it is independence between X and Y.

16. Example 2

18. Continue after a rest