1. Section 7.2.1 Means and Variances of Random Variables
AP Statistics

2. Random Variables: Mean
AP Statistics, Section 7.2, Part 1

3. Random Variables: Example
The Michigan Daily Game you pick a 3 digit number and win $500 if your number matches the number drawn.
What is the average winnings?
AP Statistics, Section 7.2, Part 1

4. Random Variables: Example
The Michigan Daily Game you pick a 3 digit number and win $500 if your number matches the number drawn.
What is the average PROFIT?
Mean = Expected Value
AP Statistics, Section 7.2, Part 1

5. Random Variables: Variance
AP Statistics, Section 7.2, Part 1

6. AP Statistics, Section 7.2, Part 1
Law of Large Numbers
Draw independent observations at random from any population with finite mean μ.
Decide how accurately you would like to estimate μ.
As the number of observations drawn increases, the mean x-bar of the observed values eventually approaches the mean μ of the population as closely as you specified and then stays that close.
AP Statistics, Section 7.2, Part 1

7. AP Statistics, Section 7.2, Part 1
Example
The distribution of the heights of all young women is close to the normal distribution with mean 64.5 inches and standard deviation 2.5 inches.
What happens if you make larger and larger samples…
AP Statistics, Section 7.2, Part 1

8. AP Statistics, Section 7.2, Part 1

9. AP Statistics, Section 7.2, Part 1
Law of Small Numbers
Most people incorrectly believe in the law of small numbers.
“Runs” of numbers, etc.
AP Statistics, Section 7.2, Part 1

10. AP Statistics, Section 7.2, Part 1
Assignment
Exercises: odd x21
AP Statistics, Section 7.2, Part 1

11. Section 7.2.2 Means and Variances of Random Variables
AP Statistics

12. AP Statistics, Section 7.2, Part 1
Rules for Means
Rule 1: The same scale change of elements of a probability distribution has the same effect on the means.
Rule 2: The mean of sum of the two distributions is equal to the sum of the means.
AP Statistics, Section 7.2, Part 1

13. AP Statistics, Section 7.2, Part 1
Rule 1 Example
A company believes that the sales of product X is as follows. X
1000
3000
5000
10,000
P(X) .1 .3 .4 .2
AP Statistics, Section 7.2, Part 1

14. AP Statistics, Section 7.2, Part 1
Rule 1 Example
If the expected profit on each sale of Product X is $2000, what is the overall expected profit?
AP Statistics, Section 7.2, Part 1

15. AP Statistics, Section 7.2, Part 1
Rule 1 Example
A company believes that the sales of product Y is as follows. Y
300
500
750
P(Y) .4 .5 .1
AP Statistics, Section 7.2, Part 1

16. AP Statistics, Section 7.2, Part 1
Rule 1 Example
If the expected profit on each sale of Product Y is $3500, what is the overall expected profit?
AP Statistics, Section 7.2, Part 1

17. AP Statistics, Section 7.2, Part 1
Rule 2 Example
What is the total expected profits combined of both Product X and Product Y?
AP Statistics, Section 7.2, Part 1

18. Rules for Variances of Independent Distributions
Only if the distributions are independent can you apply these rules…
Rule 1: If a scale change involves a multiplier b, the variance changes by the square of b.
Rule 2: The variance of sum of the two distributions is equal to the sum of the variances.
Rule 2b: The variance of difference of the two distributions is equal to the sum of the variances.
AP Statistics, Section 7.2, Part 1

19. AP Statistics, Section 7.2, Part 1
Example
The Daily 3 lottery has the following mean and variance for its payout:
What is the mean and variance of the winnings?
AP Statistics, Section 7.2, Part 1

20. AP Statistics, Section 7.2, Part 1
Example
The Daily 3 lottery has the following mean and variance for its payout:
What is the mean and variance of the payouts of playing twice?
AP Statistics, Section 7.2, Part 1

21. AP Statistics, Section 7.2, Part 1
Example
The Daily 3 lottery has the following mean and variance for its payout:
What is the mean and variance of the payouts of playing every day of the year?
AP Statistics, Section 7.2, Part 1

22. AP Statistics, Section 7.2, Part 1
Assignment
Exercises, section 7.2: all
Exercises, chapter review: all
AP Statistics, Section 7.2, Part 1