1. PRODUCT MOMENTS OF BIVARIATE RANDOM VARIABLES
8.1. Covariance of Bivariate Random Variables
Definition 8.1.
Let X and Y be any two random variables with joint density function f(x, y). The product moment of X and Y , denoted by E(XY ), is defined as

2. Cov(X, Y ) = E(XY ) − E(X)E(Y ).
Theorem 8.1.
Let X and Y be any two random variables. Then
Cov(X, Y ) = E(XY ) − E(X)E(Y ).
Proof:
Cov(X, Y ) = E((X − μX) (Y − μY ))
= E(XY − μX Y − μY X + μX μY )
= E(XY ) − μX E(Y ) − μY E(X) + μX μY
= E(XY ) − μX μY − μY μX + μX μY
= E(XY ) − μX μY
= E(XY ) − E(X)E(Y ).

3. Let X and Y be discrete random variables with joint density
Example 8.1.
Let X and Y be discrete random variables with joint density
What is the covariance "XY between X and Y .
Answer:
Cov(X, Y ) = E(XY ) − E(X)E(Y )
The marginal of X is

7. Example:8.2
Let X and Y have the joint density function
𝑓 𝑥,𝑦 = 𝑥+ 𝑦 𝑖𝑓 0<𝑥<1 𝑎𝑛𝑑 0<𝑦<1 0 𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
What is the covariance between X and Y

12. 8.2. Independence of Random Variables
Theorem 8.3.
If X and Y are independent random variables, then
E(XY ) = E(X)E(Y ).
Theorem 8.4.
If X and Y are independent random variables, then the
covariance between X and Y is always zero, that is
Cov ( X, Y ) = 0.

13. Let the random variables X and Y have the joint density
Example 8.6.
Let the random variables X and Y have the joint density
What is the covariance of X and Y ? Are the random variables X and Y independent?

14. From this table, we see that
Next, we compute the covariance between X and Y . For this we need
E(X), E(Y ) and E(XY ). The expected value of X is

15. Cov(X, Y ) = E(XY ) − E(X)E(Y ) = 0
Hence, the covariance between X and Y is given by
Cov(X, Y ) = E(XY ) − E(X)E(Y ) = 0

16. 8.4. Correlation and Independence
Theorem 8.7.
If X and Y are independent, the correlation coefficient between
X and Y is zero.

17. 8.5. Moment Generating Functions

18. What is the joint moment generating function for X and Y ?
Answer:
The joint moment generating function of X and Y is given by

19. Example 8.11.
If the joint moment generating function of the random variables X and Y is
what is the covariance of X and Y ?
Answer: