Database Reference
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 :
Search WWH ::




Custom Search