1. C HAPTER 7 D AY 4

2. R ULES FOR M EANS OF R ANDOM V ARIABLES Rule 1 : If X is a random variable and a and b are fixed numbers, then Rule 2 : If X and Y are random variables then

3. So, in other words Rule 2 says… For new μ = μ X+Y : new μ = old μ X + old μ Y For new μ = μ X-Y new μ = old μ X - old μ Y We will come back to Rule 1 in a bit!

4. R ULES FOR V ARIANCES OF R ANDOM V ARIABLES Rule 1 : If X is a random variable and a and b are fixed numbers, then Rule 2 : If X and Y are independent random variables, then The last situation is COUNTERINTUITIVE

5. So, in other words Rule 2 says… For new σ 2 = σ 2 X+Y : new σ 2 = old σ 2 X + old σ 2 Y For new σ 2 = σ 2 X-Y : new σ 2 = old σ 2 X + old σ 2 Y We will come back to Rule 1 in a bit!

6. S TANDARD D EVIATIONS There are no rules for standard deviations, BUT we can use the rules for variances to form summarize for standard deviations:

7. E XAMPLE Jim’s bowling score has an average μ X = 192 and standard deviation σ X = 7 Bob’s bowling score has an average μ Y = 184 and standard deviation σ Y = 4 If Jim and Bob form a bowling team for a fundraiser, what would be their team’s average total score? What would be the total score standard deviation? Notice, no probability models were given!

8. B ACK TO R ULE #2 FOR B OTH M EANS AND V ARIANCES … Situation 1: The original data is multiplied or divided by a number new μ = old μ*b new σ 2 = old σ 2 *b 2 Situation 2: The original data is added or subtracted by a number new μ = old μ + a new σ 2 = old σ 2 Situation 3: The original data is changed by a combo μ a + bX = a + bμ X σ 2 a + bX = b 2 * σ 2

9. L ET X = THE ORIGINAL DATA DISTRIBUTION What happens to each individual data point? New μ Z New σ 2 Z New σ Z μ X = 10σ 2 X = 9 σ X =

10. Y OU T RY !! L ET X = THE ORIGINAL DATA DISTRIBUTION What happens to each individual data point? New μ Z New σ 2 Z New σ Z μ X = 5σ 2 X = 4 σ X =

11. E XAMPLE Suppose a car salesman sells an average of 15 new cars in one month (standard deviation of 3) Their paycheck is calculated by the formula: Pay = 1000 (# cars sold) + 500 Calculate the expected monthly paycheck: μ M σ 2 M σ M Calculate the expected yearly pay: μ Y σ 2 Y σ Y