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HW- pg. 499-500 (7.38-7.42)
Ch. 7 Test MONDAY 12-16-13
www.westex.org HS, Teacher Website
12-12-13
Warm up—AP Stats
The payoff X of a $1 ticket in the Tri-State Pick 3
game is $600 with probability 1/1000 and $0 the
rest of the time. Fill in the chart below.
xi
pi
Σpi = 1
xipi
X =
(xi -  X )2pi
=
=
 X2 =
X =
Name _________________________
AP Stats
7 Random Variables
7.2 Day 3 Means and Variances of Random Variables
Date _______
Objectives
 Given  X and Y , calculate  X  Y .
2
 Given  X and Y, calculate  X  Y .
 Explain how standard deviations are calculated when combining random variables.
Think way back to the warm up
Your winnings (not average winnings) are W = ______.
The MEAN amount you win is W = _______ = _______.
On average you lose _______ per ticket. Subtracting (or adding) a fixed number __________
your ________ but not your ____________ or standard deviation of the winnings W = X – 1.
Now, let’s say you buy a $1 ticket on two different days. The payoffs X and Y on the two
tickets are _______________ because separate drawings are held each day.
Total payoff X + Y has mean:
 X  Y = _______ = __________ = ______
Because X and Y are INDEPENDENT, the variance of X + Y is:
 X2  Y = ________ = __________ = _______
The standard deviation of the total payoff is:
 X  Y = ______ = ______
which is different from the sum of the individual standard deviations, which is
_____ + _____ = _____.
CONCLUSION-***Means ALWAYS _____, Variances _____ when random variables are
_______________, and standard deviations never add!***
Why don’t variances add when random variables aren’t independent?
Assume X is the % of a family’s after-tax income that is spent and Y is the % that is saved.
When X increases Y __________ by the same amount. Though ___ and ___ may vary widely
from year to year their sum ______ is always 100% and does not _______ at all! It’s the
association (the fact that they aren’t independent) between the variables X and Y that prevents
their variances from adding.
See Examples 7.12 & 7.13 below to understand the role of independence (or lack thereof).
Combining Normal Random Variables
If a random variable is Normally distributed, we can use its mean and variance to compute
_______________. Example 7.4 (pg. 474) showed this. Now let’s talk about combining TWO
Normal random variables.
Any linear combination of _______________ Normal random variables is also Normally
distributed. The sum or difference of independent Normal random variables has a Normal
distribution.
Example 7.14
Theresa and Shannon are playing golf. Their scores vary each time they play. Theresa’s score
X has the N(110, 10) distribution, and Shannon’s score Y has the N(100, 8) distribution. If they
play independently (each of their scores has no impact on the other person’s scores), what is
the probability that Theresa will score lower than Shannon? Should it be less than 50%, why?
We need to think about the difference (X – Y) between their scores. It is Normally distributed
since each of their scores is Normally distributed.
 X  Y = _______ = __________ = ______
Because X and Y are INDEPENDENT, the variance of X + Y is:
 X2 Y = ________ = __________ = _______
The standard deviation of the difference in their scores is:
 X  Y = ______ = ______
Since the standard deviation is 12.8, X – Y has the N(10, 12.8) distribution.
P(X < Y) = P(X – Y < 0)
  X  Y   10 0  10 

= P

12.8
12.8 

= P(Z < -0.78) = _____