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Section 7.2.2
Means and Variances
of Random Variables
AP Statistics
www.toddfadoir.com/apstats
AP Statistics, Section 7.2, Part 1
2
Rules for Means


Rule 1: The same scale
change of elements of a
probability distribution
has the same effect on
the means.
Rule 2: The mean of sum
of the two distributions is
equal to the sum of the
means.
a bX  a  b X
 X Y   X  Y
AP Statistics, Section 7.2, Part 1
3
Rule 1 Example

A company believes that the sales of
product X is as follows.
X
1000
3000
5000
10,000
P(X)
.1
.3
.4
.2
 X  1000  .1  3000  .3  5000  .4  10000  .2
 X  5000 units
AP Statistics, Section 7.2, Part 1
4
Rule 1 Example

If the expected profit on each sale of
Product X is $2000, what is the overall
expected profit?
 X  1000  .1  3000  .3  5000  .4  10000  .2
 X  5000 units
02000 X  0  2000 X  10,000,000
AP Statistics, Section 7.2, Part 1
5
Rule 1 Example

A company believes that the sales of
product Y is as follows.
Y
300
500
750
P(Y)
.4
.5
.1
Y  300  .4  500  .5  750  .1
Y  445 units
AP Statistics, Section 7.2, Part 1
6
Rule 1 Example

If the expected profit on each sale of
Product Y is $3500, what is the overall
expected profit?
Y  300  .4  500  .5  750  .1
Y  445 units
03500Y  0  3500Y  1,557,500
AP Statistics, Section 7.2, Part 1
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Rule 2 Example

What is the total expected profits
combined of both Product X and
Product Y?
2000 X  10, 000, 000
3500Y  1,557,500
2000 X 3500Y  10, 000, 000  1,575,500
 11,557,500
AP Statistics, Section 7.2, Part 1
8
Rules for Variances of Independent
Distributions




Only if the distributions are
independent can you apply
these rules…
Rule 1: If a scale change
involves a multiplier b, the
variance changes by the
square of b.
Rule 2: The variance of sum of
the two distributions is equal to
the sum of the variances.
Rule 2b: The variance of
difference of the two
distributions is equal to the
sum of the variances.

2
a  bX
b 
2
2
X
 X2 Y   X2   Y2
 X2 Y   X2   Y2
AP Statistics, Section 7.2, Part 1
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Example


The Daily 3 lottery
has the following
mean and variance
for its payout:
What is the mean and
variance of the
winnings?
 X  .50
  249.75
 X  15.80
2
X
 X 1  .50
  249.75
 X 1  15.80
AP Statistics, Section 7.2, Part 1
2
X 1
10
Example


The Daily 3 lottery
has the following
mean and variance
for its payout:
What is the mean and
variance of the
payouts of playing
twice?
 X  .50
  249.75
 X  15.80
2
X
 X  X  .50  .50  1.00

 249.75  249.75
2
X X
 X  X  22.34
AP Statistics, Section 7.2, Part 1
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Example


 X  .50
The Daily 3 lottery
has the following
2
 X  249.75
mean and variance
for its payout:
 X  15.80
What is the mean and
variance of the
 X  X   X  .50  365  182.5
payouts of playing
every day of the
 X2  X  X  249.75  365  91158.75
year?
 X  X  X  301.92
AP Statistics, Section 7.2, Part 1
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Assignment
Exercises, section 7.2: 7.34-7.48 all
 Exercises, chapter review: 7.54-7.68 all

AP Statistics, Section 7.2, Part 1
14