Section Means and Variances of Random Variables AP Statistics

AP Statistics, Section 7.2, Part 1 2 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.

Example 1: We are tracking the temperature in Tracy. Let F=the degrees in Fahrenheit. The probability distribution is: Temp in degrees F P(F)

Rules for Variances AP Statistics, Section 7.2, Part 1 4 *a has no effect on the variance. When I add a to all #’s it shifts the distribution, but it’s variance(how skinny/fat) stays the same. *X and Y have no correlation p, hence p=0

Rules for Variances(continued) AP Statistics, Section 7.2, Part 1 5

6 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 7 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 8 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 9 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 10 Rule 2 Example What is the total expected profits combined of both Product X and Product Y?

AP Statistics, Section 7.2, Part 1 11 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 12 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 13 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? *Note – Variances of independent random variables add; standard deviations do not.

AP Statistics, Section 7.2, Part 1 14 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 15 Assignment

AP Statistics, Section 7.2, Part 1 16 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.

Playing roulette $1 vs $5min bet AP Statistics, Section 7.2, Part 1 17

Dealing with Temperature “change” AP Statistics, Section 7.2, Part 1 18

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“a” has no effect on varience AP Statistics, Section 7.2, Part 1 20

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AP Statistics, Section 7.2, Part 1 27 Assignment Exercises, section 7.2: all (Due Monday). Summary due Monday