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In-Depth Information
The target quantity (
11.1
)
d
thus is the quotient of two normal distributions. As
for the treatment thereof, a paper of Robert Geary [Gea30] turns out to be helpful.
It considers the expression
b þ y
a þ x
,
z ¼
ð
11
2
Þ
:
where
x
and
y
are normally distributed with expected values 0, standard deviations
α
,
β
, and correlation
r
. Under these assumptions, the expression
az b
α
t ¼
p
ð
11
3
Þ
:
2
2
z
2
2
r
αβ
z þ β
is standard normally distributed
N
(0,1) if
a
+
x
is in general nonnegative. This
assumption is easily satisfied in our case because we consider revenues.
We may apply this insight to our task at hand. From (
11.1
), it readily follows that
X
B
X
A
:
d þ
1
¼ z ¼
ð
11
:
4
Þ
With
a ¼ EX
A
,
b ¼ EX
B
,
D
2
X
A
n
A
2
α
¼
,
D
2
X
B
n
B
2
β
¼
,
r ¼
0,
we obtain (
11.2
).
By virtue of (
11.4
), we may derive the desired confidence interval for
d
. Due to
(
11.3
),
t
is normally distributed. Let
U
P
be the
p
-quantile of the standard normal
distribution for probability
p
.Then
¼
1
2
p
PU
P
t U
1
p
:
ð
Þ
The problem is symmetric and therefore it holds that
t
2
U
p
: