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AP STATISTICS
Chapter 7 – Random Variables
Section 7.2: Means and Variances of Random Variables (1)
Homework Assignment: Exercises 7.22,7.23,7.29
Name _______________________
Date __________
Period _____
A. Review Assignment: Exercises 7.2, 7.6, 7.8, 7.10, 7.12, 7.16
B. Key Vocabulary: mean of a random variable (  x ), variance of a random variable (  X2 ), expected value
of X [ E(X) ],  X   xi pi ,  X2   xi   X  pi , standard deviation of X, (  X ), law of large numbers,
2
rules for means:  a bX  a  b X ,  X Y   X  Y , rules for variances:  a2bX  b 2 X2 ,
 X2 Y   X2 Y   X2   Y2 (so long as X and Y are independent – variances add), linear combination of
independent normal random variables is also normally distributed, correlation between two independent
random variables is zero
Whoa!!!!! Some pretty intimidating looking stuff up there!!
C. Discrete Random Variable Situation: Finding mean (expected value), variance, and standard deviation
Let X be the number of heads counted when 3 coins are tossed. X is a discrete random variable.
X
P(X)
0
1/8
1
3/8
2
3/8
3
1/8
Sketch the probability histogram:
0
1
2
3
Mean (expected value) calculation:  X   xi pi
Variance calculation:  X2   xi   X  pi
2
Standard deviation calculation (square root of variance):
(Do on calculator: L1  values of X, L2  values of P(X); 1-Var Stats L1,L2)
D. Continuous Random Variable Situations
The mean and variance (or standard deviation) are provided for you in the question.
Page 422: Example 7.12 SAT Scores
SAT Math score (X)
SAT Verbal score (Y)
X = 625
Y = 590
X = 90
Y = 100
E. The Law of Large Numbers
Draw independent observations at random from any population with finite mean . Decide how accurately
you would like to estimate . As the number of observations drawn increases, the mean x of the observed
values eventually approaches the mean  of the population as closely as you specified and then stays that
close.
Example: The distribution of the heights of all young women is close to N(64.5,2.5). Pick out how close
you would like your sample mean to be to the truth (64.5). As the sample size n is increased, x  
Calculus students: Limits notation, anyone?
F. Rules for Means and Variances: If the transformation a+bx is applied to the values of a random
variable, then
 a bX  a  b X
and
 a2bX  b 2 X2
Example: Suppose the temperatures (F) of patients in a certain surgical situation are controlled to be
N(98.1,.2).
What would the mean (  C ) and variance (  C2 ) be if we transform by converting to Celsius?
F = (9/5)C + 32
C = -17.78 +(5/9)F
 C   17.78(5 / 9) F  17.78  (5 / 9)  F  17.78  (5 / 9)(98.1)  36.72
 C2   217.78(5 / 9) F  (5 / 9) 2  F2  (5 / 9) 2 .2 2  0.01234
and  C  0.01234  0.1111