Of course, variances are in squared units. The company would prefer to know stan
dard deviations, which are in dollars. The standard deviation of the payout for two inde
pendent policies is
1
299,200
=
+
546.99. But the standard deviation of the payout for a
single policy of twice the size is
1
598,400
=
+
773.56, or about 40% more.
If the company has two customers, then, it will have an expected annual total payout
of $40 with a standard deviation of about $547.
For Example
ADDING THE DISCOUNTS
RECAP:
The Valentine’s Day Lucky Lovers discount for couples averages $5.83 with
a standard deviation of $8.62. We’ve seen that if the restaurant doubles the dis
count offer for two couples dining together on a single check, they can expect to
save $11.66 with a standard deviation of $17.24. Some couples decide instead to
get separate checks and pool their two discounts.
QUESTION:
You and your amour go to this restaurant with another couple and agree
to share any benefit from this promotion. Does it matter whether you pay separately
or together?
ANSWER:
Let
X
1
and
X
2
represent the two separate discounts, and
T
the total; then
T
=
X
1
+
X
2
.
E
1
T
2
=
E
1
X
1
+
X
2
2
=
E
1
X
1
2
+
E
1
X
2
2
=
5.83
+
5.83
=
$11.66,
so the expected saving is the same either way.
The cards are reshuffled for each couple’s turn, so the discounts couples receive
are independent. It’s okay to add the variances:
Var
1
T
2
=
Var
1
X
1
+
X
2
2
=
Var
1
X
1
2
+
Var
1
X
2
2
=
8.62
2
+
8.62
2
=
148.6088
SD
1
T
2
=
1
148.6088
=
$12.19
When two couples get separate checks, there’s less variation in their total discount.
The standard deviation is $12.19, compared to $17.24 for couples who play for the
double discount on a single check.
In general,
■
The mean of the sum of two random variables is the sum of the means.
■
The mean of the difference of two random variables is the difference of the means.
■
If the random variables are independent, the variance of their sum or difference is
always the sum of the variances.
E
1
X
{
Y
2
=
E
1
X
2
{
E
1
Y
2
Var
1
X
{
Y
2
=
Var
1
X
2
+
Var
1
Y
2
Wait a minute! Is that third part correct? Do we always
add
variances? Yes.
Think about the two insurance policies. Suppose we want to know the mean and stan
dard deviation of the
difference
in payouts to the two clients. Since each policy has
an expected payout of $20, the expected difference is 20

20
=
+
0. If we also sub
tract variances, we get $0, too, and that surely doesn’t make sense. Note that if the out
comes for the two clients are independent, the difference in payouts could range from
+
10,000

+
0
=
+
10,000 to
+
0

+
10,000
=

+
10,000, a spread of $20,000. The
variability in differences increases as much as the variability in sums. If the company
has two customers, the difference in payouts has a mean of $0 and a standard deviation
of about $547 (again).
PYTHAGOREAN
THEOREM OF
STATISTICS
We often use the stan
dard deviation to measure
variability, but when we
add independent random
variables, we use their
variances. Think of the
Pythagorean Theorem. In
a right triangle (only), the
square
of the length of the
hypotenuse is the sum of
the
squares
of the lengths
of the other two sides:
c
2
=
a
2
+
b
2
.
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