1. AP Statistics Section 7.2 C Rules for Means & Variances

2. Consider the independent random variables X and Y and their probability distributions below:

3. Build a new random variable X + Y and calculate the probabilities for the values of X + Y.

4. Use your calculator to calculate the mean of the random variable X + Y. Note that the mean of the sum = ____ equals the sum of the means =______________ :

5. Use your calculator to calculate the variance of the random variable X + Y. Note that the variance of the sum equals the sum of the variances:

6. Repeat the steps above for the random variable X – Y. Verify.

7. Repeat the steps above for the random variable X – Y. Calculate the variance of the random variable X – Y. Note that the variance of the difference equals

8. Rules for Means Rule 1: If X is a random variable and a and b are constants, then.

9. Rules for Means Rule 2: If X and Y are random variables, then ______and _______

10. Rules for Variances Rule 1: If X is a random variable and a and b are constants, then ______

11. Rules for Variances Rule 2: If X and Y are independent random variables, then

12. Example: Consider two scales in a chemistry lab. Both scales give answers that vary a little in repeated weighings of the same item. For a 2 gram item, the first scale gives readings X with a mean of 2g and a standard deviation of.002g. The second scale’s readings Y have a mean of 2.001g and a standard deviation of.001g. If X and Y are independent, find the mean and standard deviation of Y – X.

13. Example: Consider two scales in a chemistry lab. Both scales give answers that vary a little in repeated weighings of the same item. For a 2 gram item, the first scale gives readings X with a mean of 2g and a standard deviation of.002g. The second scale’s readings Y have a mean of 2.001g and a standard deviation of.001g. You measure once with each scale and average the readings. Your result is Z = (X+Y)/2. Find.

14. Any linear combination of independent Normal random variables is also Normally distributed.

15. Example: Tom and George are playing in the club golf tournament. Their scores vary as they play the course repeatedly. Tom’s score X has the N(110, 10) distribution and George’s score Y has the N(100, 8) distribution. If they play independently, what is the probability that Tom will score lower than George?