Astronomical Applications Department, U.S. Naval Observatory Gaussian Distribution Page 2
$ 0RGLILHG %R[0XOOHU $OJRULWKP
7KH XVXDO DOJRULWKP IRU SURGXFLQJ D *DXVVLDQ GLVWULEXWLRQ IURP XQLIRUP UDQGRP GHYLDWHV
PDNHV XVH RI WKH %R[0XOOHU WUDQVIRUPDWLRQ :H ZLOO PRGLI\ WKH XVXDO DOJRULWKP VR WKDW WKH
%R[0XHOOHU WUDQVIRUPDWLRQ SURGXFHV WKH GHVLUHG QRQXQLWYDULDQFH *DXVVLDQ GLVWULEXWLRQ ,I
]
DQG
]
DUH LQGHSHQGHQW DQG XQLIRUPO\ GLVWULEXWHG LQ WKH UDQJH WR WKHQ FRQVLGHU
\
DQG
\
JLYHQ E\
=
\
-
OQ ] FRV
]
=
\
-
OQ ] VLQ
]
7KLV LV WKH %R[0XOOHU WUDQVIRUPDWLRQ ZKLFK FRQYHUWV D ' XQLIRUP GLVWULEXWLRQ
] ]
WR D
ELYDULDWH *DXVVLDQ GLVWULEXWLRQ
\ \
ZLWK ]HUR PHDQ DQG XQLW YDULDQFH 7KH SURRI WKDW WKLV
WUDQVIRUPDWLRQ \LHOGV WKH GHVLUHG *DXVVLDQ GLVWULEXWLRQ LV D VSHFLDO FDVH RI WKH SURRI RI WKH PRGLILHG
WUDQVIRUPDWLRQ ZKLFK LV JLYHQ EHORZ
7KH WULFN WR PDNLQJ WKH %R[0XOOHU WUDQVIRUPDWLRQ SURGXFH D *DXVVLDQ GLVWULEXWLRQ
[ [
ZLWK QRQXQLW YDULDQFH LV WR XVH WKH LQYDULDQW UHODWLRQ PHQWLRQHG DERYH 6HWWLQJ
=
[
+
\
µ
ZH KDYH
=
[
+
-
OQ ] FRV
]
µ
=
[
+
-
OQ ] VLQ
]
µ
9HULILFDWLRQ
7R VKRZ WKDW WKLV LV LQGHHG WKH GHVLUHG GLVWULEXWLRQ ZH KDYH
=
+
-
[
µ
-
[
µ
-
OQ ]
ZKLFK ZH VROYH IRU
]