Transcript The Practice of Statistics Third Edition Chapter 7

Daniel S. Yates
The Practice of Statistics
Third Edition
Chapter 7:
Random Variables
Copyright © 2008 by W. H. Freeman & Company
Combining RandomVariables
Activity 7C
Page 492
Linda Sells Cars & Trucks
• Linda’s cars
Find  X  Y
• Linda’s trucks
If she expects to earn
$350 for each car and
$400 for each truck,
find her expected
mean earnings.
Since variance is the average of squared deviations from the
mean, multiplying X by a constant b multiplies the variance
by square of the constant.
Adding a constant a to a random variable changes its mean
but does not change its variability
Because the square of -1 is 1…
Be aware… standard deviations do not generally add. Use
the rules for variance to find combined standard deviations.
Tri-State Pick 3
• Find the mean and standard deviation of the
game.
• What is the mean amount you win if each
ticket costs $1.
Tri-State Pick 3
• What is the mean, variance, and standard
deviation of the total payoff if you buy two
tickets on two different days?
• Notice variances of independent random
variances add; standard deviations do not.
SAT scores
• A college uses SAT scores as one criterion for
admission. Experience has shown that the
distribution of SAT scores among its entire
population of applicants is such that
 X  519  X  115
VerbalY Y  507  Y  111
MathX
• What is the mean and standard deviation of the
total score X + Y among students applying to this
college?
Normal calculations &
combining means
Any linear combination of independent
Normal random variables is also Normally
distributed. That is, if X and Y are
independent Normal random variables and a
and b are any fixed numbers, aX + bY is
also Normally distributed.
(Basically we can use Normal calculations with combined means and
variance as well)
A Round of Golf
Combining Random Variables
• Tom and George are playing in the club golf tournament.
Their scores vary as they play the course repeatedly.
Tom’s score X has the N(110, 10) distribution, and
George’s score Y varies from round to round according to
the N(100, 8) distribution. If they play independently,
what is the probability that Tom will score lower than
George and thus do better in the tournament?
A. Is the difference in their means Normally distributed?
A Round of Golf
Combining Random Variables
• Tom and George are playing in the club golf tournament.
Their scores vary as they play the course repeatedly.
Tom’s score X has the N(110, 10) distribution, and
George’s score Y varies from round to round according to
the N(100, 8) distribution. If they play independently,
what is the probability that Tom will score lower than
George and thus do better in the tournament?
B. Find the mean and standard deviation of the difference in
the scores.
(Identify the parameters of the distribution of X-Y.)
A Round of Golf
Combining Random Variables
• Tom and George are playing in the club golf tournament.
Their scores vary as they play the course repeatedly.
Tom’s score X has the N(110, 10) distribution, and
George’s score Y varies from round to round according to
the N(100, 8) distribution. If they play independently,
what is the probability that Tom will score lower than
George and thus do better in the tournament?
C. Given N(10, 12.8), find P(X<Y)
Another example? 7.42