Astronomical Applications Department, U.S. Naval Observatory Gaussian Distribution Page 3
=
] 0
- /
-
[
µ
0
- /
-
[
µ
DQG ZH KDYH
=
[
[
+
-
OQ ] VLQ
]
µ
+
-
OQ ] FRV
]
µ
ZKLFK ZH VROYH IRU
]
]
DUFWDQ
-
OQ ] [
- +
[ [
µ
-
=
+
OQ ] [
- +
[ [
µ
OQ ]
[
+
[
[
-
[
-
OQ ] [
- +
[ [
µ
[
+
OQ ] [
- +
[ [
µ
OQ ]
[
+
[
[
+
/
6XEVWLWXWLQJ IRU
]
DQG VLPSOLI\LQJ ZH ILQG
=
]
DUFWDQ
- +
[
µ
- +
[
µ
1RZ WKH IXQGDPHQWDO WUDQVIRUPDWLRQ ODZ RI SUREDELOLWLHV VWDWHV WKDW
=
S \ G\
S [ G[
RU
=
S \
S [
G[
G\
ZKHUH
S [ G[
LV WKH SUREDELOLW\ WKDW [ OLHV EHWZHHQ [ DQG
+
[ G[
DQG
S \ G\
LV WKH SUREDELOLW\ WKDW \ OLHV EHWZHHQ \ DQG
+
\ G\
7KH H[WHQVLRQ RI WKLV WR Q GLPHQVLRQV LV
=
S < G< S ; - ; < G<
ZKHUH
S ;
LV WKH MRLQW SUREDELOLW\ GLVWULEXWLRQ RI ;
S <
LV WKH MRLQW
SUREDELOLW\ GLVWULEXWLRQ RI <
=
;
[
[ [Q
=
<
\
\ \Q
=
G<
G\ G\ G\Q
DQG
- ; <
LV WKH -DFREL GHWHUPLQDQW )RU WZR GLPHQVLRQV
=
- ; <
GHW
\
[
\
[
\
[
\
[
+HQFH
