Download AP Stats CH7 Combining Random Variables

Combining and “Adjusting”
Discrete Random Variables
Section 7-2
Two Rules regarding means
Rule 1 for Means:
If you’re doing a linear
transformation, play it just as we did
before:
New = a + b Old
…where
◦ a is a constant you might add to every value
of the random variable, and
◦ b is a constant by which you might multiply
every value of the random variable
Example: Gain Electronics
(page 419)
X
1000
3000
5000
10,000
Probability
.1
.3
.4
.2

X represents numbers of radios for military applications.

The probabilities are the estimates by the company’s executives. For instance, the executives
think there’s a .1 probability that they’ll only sell 1000 units, a .3 probability of selling 3000
units, etc.
Question: What’s the best guess of the number of units that will
be sold next year for military applications?
Example: Gain Electronics
(page 419)
X
1000
3000
5000
10,000
Probability
.1
.3
.4
.2
Question: What’s the best guess of the number of units that will
be sold next year?
(Hint: “Best guess” = “Expectation” = “Expected Value” = Mean)
E(x) = μx = _______________
Example: Gain Electronics
(page 419)
X
1000
3000
5000
10,000
Probability
.1
.3
.4
.2
Question: If we profit $2000 for every radio, what will be our expected total
profit?
profit
= 2000(radios)
= 2000(5000)
= $10,000,000


Notice: a = 0; b = 2000. In other words, there really is an a and b. a simply happens to equal 0.
Also notice that you would get this same answer if you did indeed multiply each value of X by 2000, then
recalculate the mean.
Rule 2 for Means:
If you’re combining
two variables through addition (or
subtraction), the mean of the new
distribution is the sum (or diff.) of the
two distributions separately.
μX+Y = μX + μY
Example: Gain Electronics
(page 419)
Y
300
500
750
P(Y)
.4
.5
.1

New random variable Y represents numbers of radios for civilian applications.
Question: What’s the best guess of the total number of units that will be sold next
year? (Military and Civilian?)
μX+Y
μMil + Civ
= μX + μY
= μMil + μCiv
= 5000 + 445
= 5445 radios
Again—”Best Guess” = “Expected Value” = “Mean”
Throw it all together…
What’s the expected total income for
Gain Electronics?
 Use both rules.



E(total income) = E(military income) + E(civilian income)\
E(total income) = $2000 (5000 radios) + $3500 (445 radios)
= $11, 557, 500
◦ Notice—There’s multiplying and adding going on!
Rules for Variance

Looky here!
 new  b old
◦ Remember how when you’re
performing a linear
transformation, Standard
Deviation is only effected by
multiplicative constants?
(Chapter 1, p. 55)
◦ Well—what happens when you
square both sides?

2
 (b old )  b 
2
new
2
2
old
Rules for Variance

1. To find the standard deviation of a
combined random variable, add variances,
then take the square root.

NEVER EVER ADD STANDARD
DEVIATIONS!

2. Even when combining random variables
through subtraction, add variances.
Rules for Variance
◦ Again
◦ Example: The distribution of
random variable X is N(24, 4).
Find the Var(X) if you triple
every value.

2
 (b old )  b 
2
new
 2 new  b 2 2old
 2 new  32 42  96
2
2
old
In your textbook—Go to p. 428
Do problem 7.46
What about combining more than
two Random Variables?
Mean: Add all the means
together.
 Standard Deviation: Add all
the variances together, then
take the square root of the
sum.

Example:

Team of 4 students
Teams of four
students are
randomly selected
from a class whose
grade distribution
is in the table.
4
3
2
1
.2
.4
.2
.2

Find the total mean
and standard
deviation for
combinations of
four students
Example:
Teams of 4 students
Mean: Take mean
for the singlestudent distribution
(2.6), and add it
four times.
 µ = 10.4

4
3
2
1
.2
.4
.2
.2

Standard
Deviation: Add
the variance four
times, then take the
square root.
  1.04  1.04  1.04  1.04  2.04