1. Rules for Means and Variances

2. Rules for Means: Rule 1: If X is a random variable and a and b are constants, then If we add a constant a to every value of a random variable, then its mean will have a added, as well. If we multiply every value of a random variable by a constant, then its mean will be multiplied by that constant, also.

3. X 0 1 2 3 4 P(X).1.3.2.3.1 3+1.5X 3 4.5 67.5 9 P(3+1.5X).1.3.2.3.1 We found in an earlier exercise that  X =2. Suppose we now find  3+1.5X. By both direct calculation and applying the rule for means, we find that  X =6. ( ) Check this for yourself. Let’s try an example. Let X be the random variable shown:

4. Rules for Means: Rule 2: If X and Y are random variables, then When we find the sum of two random variables, the mean is the sum of the individual random variables.

5. Rules for Means: Rule 2a: If X and Y are random variables, then When we find the difference of two random variables, the mean is the difference of the individual random variables.

6. Now for some examples, suppose the mean of X is 5 and the mean of Y is 3. To find  X+Y we add  X and  Y.So,  X+Y =5+3=8. To find  X-Y we subtract  X and  Y. So,  X-Y =5-3=2. That’s not so hard!

7. Rules for Variances: Rule 1: If X is a random variable and a and b are constants, then If we add a constant a to every value of a random variable, then its variance is unchanged. If we multiply every value of a random variable by a constant, then its variance will be multiplied by the square of that constant.

8. X 0 1 23 4 P(X).1.3.2.3.1 3+1.5X 3 4.5 67.5 9 P(3+1.5X).1.3.2.3.1 We found in an earlier exercise that. By both direct calculation and applying the rule for variances, we find that Check this for yourself. Let’s try an example. Suppose we now find. ( ).

9. Rules for Variances: Rule 2: If X and Y are independent random variables, then When we find the sum of two random variables, the variance is the sum of the individual variances of the random variables, provided they are independent.

10. Rules for Variances: Rule 2a: If X and Y are independent random variables, then When we find the difference of two random variables, the variance is the sum of the individual variances of the random variables, provided they are independent. Note that we still add, not find the difference!!

11. Rules for Variances: Rule 3: If X and Y are random variables with correlation , then When we find the sum of two random variables, the variance is the sum of the individual variances of the random variables plus the additional term 2  X  Y.

12. Rules for Variances: Rule 3a: If X and Y are random variables with correlation , then When we find the difference of two random variables, the variance is the sum of the individual variances of the random variables plus the additional term 2  X  Y. Note that we did not subtract, but added the variances!!

13. Now for some examples, suppose X and Y are independent, and the standard deviation of X is 2.4 and the standard deviation of Y is 1.7. To find So, To find Notice that they are the same, whether we add or subtract the random variables. So, we addand we addand

14. Now for some examples: suppose X and Y are random variables with a correlation of.7, and the standard deviation of X is 2.4 and the standard deviation of Y is 1.7. To find So, To find Notice that they are the same, whether we add or subtract the random variables. So, we addand we addand

15. These rules for combinations of random variables are extremely important. They will not be provided on your formula sheet!

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