Areas under the normal curve
In order to use the normal curve to approximate discrete probabilities we have to be able to
calculate values of
But any table we use can only store values for a finite range of . I have chosen somewhat
arbitrarily. What to do for !" ?
There is a useful trick. First of all, we know that
$#
% so
(
% *)
1
and in our case
9
But also
./0%.
1
3
34
#
$#
,
%3 #
'& +-,
But integration by parts tells us that
21
'& ,
8#
/
5 .6
7
#
& , 8
/
.
/
#
8#
$#
,
5 .
.=< %
;: 7 ;>
#
8
,
.?< :
@>
9 :
Therefore
$#
,
,
/
/ . 6BA :
D
: C
To summarize, we can use this inequality as an estimate:
% ,
A :DC
9
#
1
/%
9
A C
For example when EFG :
HJIKHHHHHH%LM
GGON
Q P LRTS$U
V
HJI_HHHHHH%L`%ab=I
WX.YZ[\]%^ N
This process can be continued, and leads to the fact that the integral lies between any two
successive partial sums of the series
b
P
WXcdZ[\fe Eh
g
P
E ikj
.
E
V
bml
G
En2j
g
bol
G
E p
.
l'a
lqlqldr*I
g
This series is rather strange. It does not converge, but the first few terms will decrease in
magnitude, and it may therefore be used for computation.
When EsFt , for example, we get for the series
u
wdvJI_HW HXHcdHZM[b\ M%a
H ^;I_v L HHH
J
HJI_wqL v HH
HJ
G
G
HJIKH
b
HJI_Lb
HJI_HHHH%aM L
Gx
tt
x
LP
HJI_Lb%ab
HJI_HHHH%a`
g HJIKHH%P L`b
x
b
HJIKHHH`%L
HJI_Lb
HJI_HHHH%a`bH P
xt
HJI_Lb M
HJI_HHHH%a` b
g HJIKHHH H
t
t
x
t
HJIKHHH%Lb
HJI_Lb
HJI_HHHH%a`b
G
xx
x
HJI_Lb a
HJI_HHHH%a`
g HJIKHHH
x
G
x
t
a
HJIKHHH P b
HJI_Lb
HJI_HHHH%a`b%a P
xx
M
HJI_Lb a
HJI_HHHH%a`
g HJIKHHH P L
x
t
`
HJIKHHH P b
HJI_Lb
HJI_HHHH%a`b%a P
xx
H
HJI_Lb a
HJI_HHHH%a` L
g HJIKHHH P
G
x
t
P
P
The series starts to diverge here, and the most accurate possible value you can get in this way
HJI_HHHH%a`b%a
HJIKHHHH%a`
is ’is between
and
t .
P
H
2