Rules for Means and Variances

Rules for Means and Variances Target Goal: I can find the mean and standard deviation of the sum or difference of independent random variables. I can determine if two random variables are independent. I can probabilities of independent Normal random variables. 6.2b h.w: pg 378: 49, 51, 57 – 59, 63

Rules for Means • Rule #1 • If X is a random variable and a and b are fixed numbers, then • If X and Y are random variables, then

Warm up: • Suppose the equation Y = 20 + 10X converts a PSAT math score, X, into an SAT math score, Y. Suppose the average PSAT math score is 48. What is the average SAT math score?

Example: 5 years later • Let represent the average SAT math score. • Let represent the average SAT verbal score.

represents the average combined SAT score. Then is the average combined total SAT score.

Rules for Variances • (Addition rule not always true for variances.) • Take X to be the % of a family’s after tax income that is spent. • Take Y to be the % that is saved.

If X goes down , Y goes up but the sum of X + Y always equals 100% and does not vary at all. • It is the association between X and Y that prevents their variances from adding.

Independent • If random variables are independent, the association between them is ruled out and their variances will add.

Rules for Variances If X is a random variable and a and b are fixed numbers, then • Rule #1

Rule #2 • If X and Y are independent random variables, then

Note: • The variances of independent variables add but their standard deviations do not! σX+Y = sqrt (σ2X + σ2Y),not σX+Y = σX + σY • Also, the variance of the differenceis the sum of the variances. • (b/c the square of -1 is 1, pg. 421).

Example: • Suppose the equation Y = 20 + 10X converts a PSAT math score, X, into an SAT math score, Y. Suppose the standard deviation for the PSAT math score is 1.5 points.

What is the standard deviation for the SAT math score?

Standard Deviation: σX • We prefer σXas a measure of variability. • Use the rules for variance and then take the square root of.

Example: Winning the Lottery The payoff X of a $1 ticket in the Tri-State Pick 3 game is $500 with probability 1/1000 and $0 the rest of the time. Here is the combined calculation of mean and variance.

Calculation of mean and variance • So, the standard deviation σx = sqrt 249.75 = $15.80 • Games of chance have high σx.

Winnings: W = X - $1 μW= μX – 1, “payoff – cost” • 0.5– 1 = -$0.50 • Players lose money on average. The standard deviation σWof W = X -1 is the same as σW of X. • Subtracting a fixed number affects the mean not the variance.

Buy tickets two days in a row: Payoff: X + Y Find the mean and standard deviation. • Mean • μX+Y = μX + μY = $0.50 + $0.50 = $1.00

Standard Deviation X and Y are independent so, σ2X+Y = σ2X + σ2Y = 249.75 + 249.75 = 499.50 σX+Y = sqrt 499.5 Standard dev. of the total payoff = $22.35

What does this mean? • Your mean payoff for a year is 0.50 x 365 = $182.50 • Your cost to play is $365.00 • The state mean winnings is 365 – 182.50 = $182.50

Combining Normal Random Variables Linear combinations of independent random variables are also normally distributed. • If X and Y are independent and, • a and b are fixed numbers, • aX + bY is normally distributed • Find μ and σ using the rules.

Example: A round of Golf Tom and George are playing in the club golf tournament. Their scores vary as they play the repeatedly. • Tom’s score is N(110,10) • George’s score is N(100,8) If they play independently, what is the probability Tom will score lower than George?

The difference in their scores X – Y is normally distributed with: μX-Y = μX – μY = 110 – 100 = 10 σ2X-Y = σ2X + σ2Y = 102 + 82 = 164 σX-Y = 12.8

So, X – Y has the N(10, 12.8) distribution. Standardize to compute the probability. • P(X<Y) = P(X – Y < 0) • = P (X-Y) – 10 < 0 - 10 12.8 12.8 • = P(Z < -0.78) • Use Table A or calculator;

2ndVARS:normalcdf(-E99,-.78)= 0.2177 Conclusion • Although George’s score is 10 strokes lower on the average, Tom will have the lower score in about one of every five matches.

Rules for Means and Variances