Section 7.2 P1 Means and Variances of Random Variables AP Statistics

AP Statistics, Section 7.2, Part 1 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 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 7 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 8 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 9

10 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 11

AP Statistics, Section 7.2, Part 1 12 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 Rule 1 Example A company believes that the sales of product X is as follows. X ,000 P(X)

AP Statistics, Section 7.2, Part 1 14 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 Rule 1 Example A company believes that the sales of product Y is as follows. Y P(Y).4.5.1

AP Statistics, Section 7.2, Part 1 16 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 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 the 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 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 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 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 General Rules for Variances Here are the rules if the events are not independent (ρ=population correlation) 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.