Joint Probability Distributions
In general, if X and Y are two random variables, the probability distribution that defines their simultaneous behavior is called a joint probability distribution.
Note:
If X and Y are 2 discrete random variables, this distribution can be described with a joint probability mass function. If X and Y are continuous, this distribution can be described with a joint probability density function.
In the discrete case
A
in the continuous case,
Example:
Suppose we have the following joint mass function
| -2 | 0 | 5 |
1 | 0.15 | K | 0.20 |
3 | 0.20 | 0.05 | 0.15 |
Find the value of k?
Y
X
Answer:
Example:
Suppose we have the following joint density function
Answer:
The marginal distributions
Example:
Suppose we have the following joint mass function
| -2 | 0 | 5 |
1 | 0.15 | 0.25 | 0.20 |
3 | 0.20 | 0.05 | 0.15 |
Find the marginal distributions of X and Y?
Y
X
Answer:
Sum | 5 | 0 | -2 | |
0.6 | 0.20 | 0.25 | 0.15 | 1 |
0.4 | 0.15 | 0.05 | 0.20 | 3 |
1 | 0.35 | 0.30 | 0.35 | Sum |
Y
X
So
The marginal distribution of X
The marginal distribution of Y
Sum | 3 | 1 | |
1 | 0.4 | 0.6 | |
Sum | 5 | 0 | -2 | |
1 | 0.35 | 0.30 | 0.35 | |
Example:
Suppose we have the following joint density function
Find the value of c ?
Find the marginal distributions of X and Y?
Answer:
conditional probability distribution
Example:
Solution:
Statistical Independence
Example:
Suppose we have the following joint distribution
Prove that X and Y are independent?
Notes:�if X and Y are independent, then
Example:
Suppose we have the following joint distribution
Find:
Solution: