1. Expected values, covariance, and correlation

2. Expectation of a function of random variables
Let X and Y be jointly distributed rv’s with pmf or pdf f(x,y). The expected value E[h(X,Y)] of a function of the random variables is

3. Example (the nut company)
Recall that X=weight of almonds and Y=weight of cashews. The joint density is

4. Example (continued)
If 1 lb. of almonds cost the company $1, 1 lb. of cashews cost them $1.50, and 1 lb. of peanuts costs $.50, then the total cost of the contents is The expected total cost is

5. Covariance
It is frequently of interest to assess how strongly two rv’s are related to one another. The covariance between two random variables is defined by

6. Properties of covariance
The covariance is the expected product of the deviations of the rv’s from their means.
Note that
If X and Y increase together, the covariance is positive, it is negative if Y decreases when X increases, and zero if those two effects perfectly cancel each other.
The covariance depends on both the possible values of the rv’s and their probabilities.

7. Easier way to compute covariance
The following formula gives a shortcut for computing covariance:

8. Example: insurance deductibles
For the joint and marginal densities:
100
200
.20
.10 .5
250
.05
.15
.30
.25
.50

9. Example: mixed nuts
The joint and marginal densities:

10. The problem with covariance as a measure
Covariance depends on the scale of the variables.
For example, in the deductible example, if we measure X and Y in units of hundreds, Cov(X,Y)=(1875)(.01)(.01)=.1875.
This problem is avoided by scaling the covariance.

11. Correlation
The correlation coefficient between X and Y is defined as

12. Covariance for examples
For the insurance deductible example,
For the mixed nuts example,

13. Proposition If a and c are either both positive or both negative, .
For any two rv’s X and Y,
By statement 1, correlation is not affected by linear changes in the units of measurement.
By statement 2, the strongest positive correlation is 1 and the strongest negative correlation is -1.

14. Proposition If X and Y are independent, then , but
does not imply that they are independent.
or if and only if for some numbers a and b with
Thus measures linear relationships. Variables can be dependent without having a linear relationship.

15. Example Let for each of the four points
. Then (check) but the variables are not independent. (What is their relationship?)