# Rules for Means and Variances

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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 = X 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 difference is the sum of the variances. (b/c the square of -1 is 1, pg. 421).

Example: Suppose the equation Y = X 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 σX as 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
xi pi xi pi (xi – μx) 2pi 0.999 (0-0.5) ) = 500 0.001 0.5 ( ) 2 (0.001) = μx(mean payoff) = σ2x = 249.75 So, the standard deviation σx = sqrt = \$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 σW of 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 = \$1.00

Standard Deviation X and Y are independent so, σ2X+Y = σ2X + σ2Y
= = 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

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 = = 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 < = P(Z < -0.78) Use Table A or calculator;

2nd VARS: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.