1. Additional notes on random variables

2. Law of Large Numbers
The average results of many independent observations are stable and predictable.
The average of the values of X observed in many trials must approach .   
Similar to probability…. In the long run, the proportion of outcomes taking any value gets close to the probability of that value, and the average outcome gets close to the distribution mean.
It assures us that statistical estimation will be accurate if we can afford enough observations.
Note: Insurance companies rely on this.

3. Rules for means
If X is a random variable and a and b are fixed number, then
If X and Y are random variables, then
See examples on Page 327

4. Rules for Variances
If X is a random variable and a and b are fixed numbers, then
If X and Y are independent random variables, then
Addition Rule for variances of independent random variables.
If X and Y have correlation , then
General Addition Rule for variances of random variables.