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Day 40 Agenda:
Quiz 6.1 & 6.2 --- 30 minutes
Advanced Placement Statistics
Section 7.2: Means & Variance of a
Random Variable
EQ: How do you calculate the mean,
standard deviation, and variance of a
random variable?
Part I: Means and Variance of
Random Variables
Recall: Means and Variance of Samples
mean of a data set
variance of a data set
standard deviation of a data set
But do you ever have to calculate
these by hand?
Recall: Statistics
Samples
Ex. You and I play a betting game:
We flip a coin and if it lands heads, I give you a
dollar, and if it lands tails, you give me a dollar.
On average, how much am I expected to win
or lose?
RECALL: Expected Value
RECALL: Expected Value
1
1
expected winnings =  $1   $1 
2
2
lose (-$1)
win ($1)
half the time half the time
On average, in the long run, I can expect to
win $0 in this game.
x
mean of DRVx
“expected value” or “weighted average”

2
x
variance of DRVx

x
standard deviation
of DRVx
NOT ON YOUR
FORMULA SHEET!
Why are we using μ and σ for DRV mean and
standard deviation?
DRV are FINITE values and represent every value
that COULD possibly occur for the random
variable X. Therefore they are parameters.
Ex. Given the following probability
distribution:
a) Find P(X = 3). P(X = 3) = .1
Why or why not? NO, it’s
weighted data.
E(X) = μx = 2(.15) + 3(.1) + 4(.2) + 5(.2) + 6 (.35) = 4.5
On average, in the long run, the expected value for this
distribution is 4.5 (whatever the units are).
Ex. Given the following probability
distribution:
.1
x 
2  4.52 .15  3  4.52 .1  4  4.52 .2  5  4.52 .2  6  4.52 .35
σx = 1.43
Ex. Given the following probability
distribution:
.1
How can you use lists in your graphing calculator to find
expected value?
X  L1
P(X)  L2
Use 1st Var Stat
What
are
the
default
Compare the summary statistics to our values for μx and σx .
lists for 1st Var Stat ?
What do you notice about Sx? Explain.
Do p. 486 #23 (will refer back to ex
on notes 7.1)
number of girls in a randomly selected family of 3 children
E(X) = μx = 0(.125) + 1(.375) + 2(.375) + 3(.125) = 1.5
 x  0  1.52 .125  1  1.52 .375  2  1.52 .375  3  1.52 .125  0.866
Law of Large Numbers --- the result of
performing the same experiment a large
number of times. According to the law,
the “long-run average” of the results
obtained from a large number of trials
should be close to the expected value,
and will tend to become closer as more
trials are performed.
What does the proportion
of heads begin to
approach as the number
of trials gets closer to
1000?
How large is “large enough”?
More variability
in DRV
More trials needed
to reach µ
Assignment: p. 486 #24, 26, 28
Part II: Transformations of Random Variables
RECALL: Linear Transformations in Algebra
f ( x)  x
original values for x and y
f ( x)  8 x
a value of x must be
multiplied by 8 to obtain
the original y value
f ( x)  5 x  2
each value of x is
multiplied by 5 and
decreased by 2 to
obtain the original y value
Ex. Given this sample for a random variable X
X = {12, 14, 16}
NOTE: NOT WEIGHTED DATA!!
E(X) = μx = 12+14+16 = 14
3
 2x
2
2
2

12  14  14  14  16  14

4
x  4  2
2
Multiply random variable X by 4 and increase
by 3. Call this random variable Y.
3
4
51
59
67
E(X) = μy = 51+59+67 = 59
3


51  59  59  59  67  59

2
2
y
 y  64  8
2
2
2
 64
Rules for Means and Variances of
Transformed Random Variables:
Refer to the previous example.
WHAT IS
MISSING?
“a” don’t PLAY
Refer to the previous example.
WHAT IS
MISSING?
“a” don’t PLAY
Refer to the previous example.
NOTE: The standard deviation of 2X and -2X
are both equal to 2σx because this is the
square root of 4σ2x .
Rules for Means and Variances of
Combined Random Variables:
Combining these random variables
means X + Y
Example for combining random variables:
If they work on separate jobs, how many
hours, on average, will they bill for
completing both jobs?
40 hr. + 35 hr. = 75 hr.
What is the variance and standard
deviation for completing both jobs?
5 hr. + 4 hr. = 9 hr.
What is the variance and standard
deviation for completing both jobs?
5 hr. + 4 hr. = 9 hr.
5  4  9  3 hr.
REMEMBER:
 X Y  
2
X
 
3 5 4
2
Y
TRANSFORMING DOES NOT MEAN COMBINING
Assignment: p. 486 #24, 26, 28
THQ#4 due Wednesday at
the beginning of class.
We need to finish the second free
response question for Media Center Ch 6.