# Statistics 7.2.2 by z8OBCl

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```									Section 7.2.2
Means and Variances
of Random Variables
AP Statistics
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
a bX  a  b X
 X Y   X  Y
the means.
   Rule 2: The mean of sum
of the two distributions is
equal to the sum of the
means.

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   7
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                                   2
a  bX   b 
2    2
X
involves a multiplier b, the
variance changes by the
square of b.                                          X Y   X   Y2
2        2

   Rule 2: The variance of sum of
the two distributions is equal to                     X Y   X   Y
2        2     2
the sum of the variances.
   Rule 2b: The variance of
difference of the two
distributions is equal to the
sum of the variances.

AP Statistics, Section 7.2, Part 1                       9
Example
   The Daily 3 lottery                                 X  .50
has the following                                249.75
2
X
mean and variance
for its payout:                                 X  15.80
   What is the mean and
 X 1  .50
variance of the
winnings?                                        249.75
2
X 1

 X 1  15.80

AP Statistics, Section 7.2, Part 1                   10
Example
   The Daily 3 lottery                                 X  .50
has the following                                249.75
2
X
mean and variance
for its payout:                                 X  15.80
   What is the mean and                 X  X  .50  .50  1.00
variance of the
payouts of playing                 X  X  249.75  249.75
2

twice?
 X  X  22.34

AP Statistics, Section 7.2, Part 1                 11
Example
   The Daily 3 lottery                     X  .50
has the following
mean and variance                   X  249.75
2

for its payout:                       X  15.80
   What is the mean and
variance of the
payouts of playing        X  X   X  .50  365  182.5
every day of the     X  X  X  249.75  365  91158.75
2

year?
 X  X  X  301.92

AP Statistics, Section 7.2, Part 1   12
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

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