15.07.2013 Views

Statistics: Basic Concepts - Chandra X-Ray Observatory (CXC)

Statistics: Basic Concepts - Chandra X-Ray Observatory (CXC)

Statistics: Basic Concepts - Chandra X-Ray Observatory (CXC)

SHOW MORE
SHOW LESS

You also want an ePaper? Increase the reach of your titles

YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.

3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

!"#"$%"$&%'<br />

(#%$&)*+,&-."%<br />

!"#$%&'(#)(*("+,-.%<br />

/%01%023')($4-+"(%"&5#"$#0&6+0&!-$0+748-(9-<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

/012345<br />

• :+$(1%$(+";&,48&2+&,#&"##2&-$%$(-$(9-<<br />

• =0+>%>(?($(#-@A(-$0(>B$(+"-<br />

• =+(--+"&C(.#?(4++2<br />

• =%0%)#$#0&D-$()%$(+"<br />

• '$%$(-$(9%?&E--B#-<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

678)9+):-),--9)!"#"$%"$&%;<br />

• /+,&2+&,#&$%.#&2#9(-(+"-&("&'9(#"9#<<br />

F++?-;&("-$0B)#"$-G&2%$%&9+??#9$(+"-G&0#2B9$(+"G&<br />

9?%--(6(9%$(+"-&H&$++?-&%"2&$#94"(IB#-<br />

A#9(-(+"-;&(-&$4(-&487+$4#-(-&9+00#9$


!"#H-%)$,)I%"J+,+K8)5L.-J$K-,"%<br />

!"#H- _+: 5L#K.Q- *+,%$9-J#"$+,%<br />

/(!5MN5 *#J-OPQQ8 5L.-J$K-,")9-%$H,R 67#";)4PKS-J)<br />

-L.+%PJ-)"$K- )T!U +O)+SV-&"%R)18.-;)T!U<br />

M5W0*5 IQH+J$"7K% &#Q$SJ#"$+,)O$Q-% 9#"#)XP#Q$"8<br />

Y5RMZ[RIM[R\![)T!U !$H,#Q]"+]4+$%-)T!U)<br />

))<br />

I4I2^!5) \#J#K-"-J 3,"-,%$"8R).+%$"$+,% [J-XP-,"$%"<br />

5%"$K#"$+,R ))T!U (#8-%$#,;<br />

_8.+"7-%$%<br />

"-%"$,H T!U))))))))))))))))))))))))))))))))))))))))))))))T!U<br />

*/4*20W5)))))))_8.+"7-%$% W$%"J$SP"$+,)"-%"%R (-Q$`#SQ-R<br />

"-%"$,H )T!U) *+JJ-Q#"$+,%)T!U M-.-#"#SQ-R<br />

0,9-J%"#,9#SQ-;)T!U<br />

M5[25*1 *#J-OPQQ8 Z$%%$+,)#&7$-`-9; 17-),-L")<br />

I)S-""-J):#8;<br />

6-),--9)K+J-)9#"#a<br />

T!U<br />

/S%-J`#"$+,%)T!U<br />

3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

J%??&M&N#".("-&OPQQRS<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


$3")"*1"*4&(-&%&IB%"$($8&$4%$&-B))%0(T#-&2%$%<br />

3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

'$%$(-$(9-&%0#&9+)>("%$(+"-&+6&2%$%&$4%$&2+&<br />

"+$&2#7#"2&+"&B"."+,"&7%0%)#$#0-;<br />

:#%"G&%1#0%*#-&60+)&)B?$(7?#&#L7#0()#"$-&<br />

#$9U<br />

Eb)I%"J+,+K-J%)&#,,+")#`+$9)!"#"$%"$&%<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

\J+S#S$Q$"8<br />

VB)#0(9%?&6+0)%?(T%$(+"&+6&<br />

+B0&2#*0##&+6&>#?(#6U&<br />

=><br />

2#.Q#&-).J$,&$.Q-)+O)$,9$OO-J-,&-'<br />

))<br />

IQQ)-`-,"%)7#`-)-XP#Q).J+S#S$Q$"8<br />

)<br />

DL%)7?#&Z;<br />

Z@X&(-&$4#&70+>%>(?($8&+6&$40+,("*&%&X&<br />

,($4&Z&0+??&+6&$4#&2(9#&(01&$4#&2(9#&9%"&<br />

>#&>(%-#2[&\]&&"##2&$+&9%?9B?%$#&$4#&<br />

70+>%>(?($8&+6&#%94&6%9#<br />

&<br />

VB)>#0&+6&6%1+0%>?#&#1#"$-<br />

&&&F+$%?&"B)>#0&+6&&#1#"$-<br />

DL%)7?#&P;&<br />

W-#&2%$%&$+&9%?9B?%$#&70+>%>(?($8G&<br />

$4B-&$4#&70+>%>(?($8&+6&%&9?+B28&<br />

+>-#01("*&0B";<br />

"B)>#0&+6&9?+B28&"(*4$-&?%-$&8#%0&&&&<br />

RXY&2%8-<br />

E--B#-;<br />

• ?()($#2&2%$%<br />

• "+$&%??&"(*4$-&%0#&'B:),,/$,*A',/&$+&<br />

>#&9?+B28<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

\J+.-J"$-%)+O)\J+S#S$Q$"8<br />

^+0)%?(T#&$4#&_)#%-B0#&+6&>#?(#6`;<br />

!GaG5&H&$40##&#1#"$-&%"2&,#&"##2&$+&)#%-B0#&<br />

4+,&-$0+"*?8&,#&$4(".&#%94&(-&?(.#?8&$+&4%77#"&<br />

%"2&%77?8&$4#&0B?#;<br />

3O)I)$%)K+J-)Q$c-Q8)"7#,)(R)#,9)()$%)K+J-)<br />

Q$c-Q8)"7#,)*R)"7-,)I)$%)K+J-)Q$c-Q8)"7#,)*U<br />

e+QK+H+J+`)#L$+K%)f)<br />

))))))))))))))[P,9#"$+,)+O)"7-)17-+J8)+O)\J+S#S$Q$"8<br />

• &!"8&0%"2+)&#1#"$&!&4%-&%&70+>%>(?($8&<br />

&&&&&&&&&.J+STIU)S-":--,)?)#,9)=<br />

• F4#&-B0#&#1#"$&.J+STIU)E)=<br />

• E6&!&%"2&a&%0#&#L9?B-(1#&OId(E?UR))2(-K+("$)#1#"$-&$4#"&&<br />

&&&&&&&&&&&&.J+STI!(U)E).J+STIU)A.J+ST(U<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


*+,9$"$+,#Q$"8)#,9)3,9-.-,9-,&-<br />

3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

&!&%"2&a&#1#"$-&%0#&$,9-.-,9-,"&(6&$4#&70+>%>(?($8&+6&<br />

+"#&(-&B"%66#9$#2&>8&,4%$&,#&."+,&%>+B$&$4#&+$4#0;<br />

70+>O!&%"2&aS\70+>O!S70+>OaS<br />

E6&$4#&70+>%>(?($8&+6&!&2#7#"2-&+"&,4%$&,#&."+,&%>+B$&a& &&<br />

&!&*(1#"&a&&&&\]&&&+,9$"$+,#Q).J+S#S$Q$"8<br />

))))))))))))))))))))))))70+>O!&%"2&aS<br />

))))))))))))))))))70+>O!baSE<br />

)))&70+>OaS<br />

E6&!&%"2&a&%0#&("2#7#"2#"$&\]&70+>O!baS\70+>O!S<br />

E6&$4#0#&%0#&-#1#0%?&7+--(>(?($(#-&6+0&#1#"$&a&Oa Z G&a P UUUUS<br />

&&&70+>O!S&\&c70+>O!ba ( S&70+>Oa ( S<br />

!&H&7%0%)#$#0&+6&("$#0#-$<br />

a ( &H&"+$&+6&("$#0#-$G&("-$0B)#"$%?&7%0%)#$#0-G&>%9.*0+B"2&<br />

70+>Oa ( S&3&(6&."+,"&,#&9%"&-B)&O+0&("$#*0%$#S&3$C)(8*&),*D'&<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

(#8-%g)17-+J-K<br />

a%8#-d&F4#+0#)&(-&2#0(1#2&>8&#IB%$("*;<br />

70+>O!&%"2&aS&\&70+>&Oa&%"2&!S<br />

&&&&&&&&&&&&&)))).J+S)TIh(U).J+ST(U<br />

&&&&&&&&&.J+ST(hIU)E<br />

&&&&&&&&&&&&&&&&&&&&&&&&.J+STIU<br />

e(1#-&$4#&fB?#&6+0&("2B9$(+";&&<br />

&&&$4#&2%$%G&$4#&#1#"$&!G&&%0#&-B99##2("*&aG&$4#&-$%$#&+6&>#?(#6&70#9##2("*&<br />

&&&$4#&#L7#0()#"$U<br />

&70+>OaS&H&.J$+J).J+S#S$Q$"8&,4(94&,(??&>#&)+2(6(#2&>8&#L7#0(#"9#<br />

&70+>O!baS&H&Q$c-Q$7++9)<br />

)70+>Oab!S&f).+%"-J$+J).J+S#S$Q$"8)H&$4#&-$%$#&+6&>#?(#6&%6$#0&$4#&<br />

2%$%&4%1#&>##"&%"%?8T#2<br />

&70+>O!S&H&"+0)%?(T%$(+"<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

5L#K.Q-<br />

!&>+L&,($4&9+?+0#2&>%??-;&<br />

&&&&&&&&&&,4%$&(-&$4#&9+"$#"$&+6&$4#&>+L<<br />

&&&70+>O9+"$#"$&+6&$4#&>+L&b&2%$%S&g&70+>O2%$%&b&9+"$#"$&+6&$4#&>+LS<br />

DL7#0()#"$;<br />

&V&0#2&>%??-&<br />

& &:&,4($#&>%??-<br />

& &Vh:&\&ZQ&$+$%?G&."+,"<br />

&A0%,&Y&$()#-&O7B$$("*&>%9.S&OFS&%"2<br />

&*#$&R&0#2&>%??-&OfS<br />

&E+@$2)&/$('7$6),,1$)('$*&$"#'$6+FG<br />

&:+2#?&O+B0&487+$4#-(-S&\]&&&&&&&&&&&&&&&&V<br />

70+>OfS&\&<br />

&&&&&&Vh:<br />

&<br />

&&&&&&&&&&<br />

&&C(.#?(4++2&\&O&&&S&70+>OfS f& 70+>O:S F3f !<br />

&<br />

"<br />

F\Y<br />

F\YQ<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

5L#K.Q-<br />

J4%$&(-&$4#&60%9$(+"&+6&$4#&&&&&&&<br />

B"+>-9B0#2&IB%-%0-<<br />

W-#&"#,&'7($T#0&+>-#01%$(+"-<br />

I&H&IB%-%0&60%9$(+"<br />

&&&&&&&F87#3Z&IB%-%0-&&&&&&&&&VZ<br />

I\&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&\<br />

&&&&&&F87#3Z&h&F87#3P&&&&VZhVP&&<br />

iVZ]&3&"B)>#0&+6&F87#3Z&I-+<br />

iVP]&3&"B)>#0&+6&F87#3P&I-+<br />

Z@&$%.#&=+(--+"&?(.#?(4++2&,($4&$4#&<br />

)#%"&iVP]&\&OZ3ISiVZ]@I<br />

P@&#1%?B%$#&?(.#?(4++2&%$&#%94&I&%"2&VZ<br />

R@&("$#*0%$#&=OVZbIS=OVZS&+1#0&VZ<br />

[J+K)#).#.-J)S8)Z#J"$,-D]<br />

!#,%$HJ-)-")#Q).PSQ$%7-9)$,)<br />

IPH)CR)i??>)$%%P-)+O&H)":('<br />

=+-$#0(+0&=0+>%>(?($8&2(-$0(>B$(+"&6+0&$4#&<br />

IB%-%0&60%9$(+"<br />

YhX&I-+<br />

F+0B-&<br />

:+2#?-<br />

E"9?B2#&<br />

+"?8&Y&I-+<br />

.TXh9#"#Rj"8.-]=)X%+kU)E).T9#"#hXRj"8.-]=)X%+kU<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

\J+S#S$Q$"8)W$%"J$SP"$+,%<br />

\J+S#S$Q$"8))$%)&JP&$#Q)$,)9-&$%$+,).J+&-%%')<br />

5L#K.Q-')<br />

)<br />

2$K$"-9)9#"#)8$-Q9%)+,Q8).#J"$#Q)$9-#)#S+P")"7-)Q$,-)<br />

:$9"7)$,)"7-)%.-&"JPKF)6-)&#,)+,Q8)#%%$H,)"7-)<br />

.J+S#S$Q$"8)"+)"7-)J#,H-)+O)"7-)Q$,-):$9"7)J+PH7Q8)<br />

K#"&7$,H)"7$%).#JK-"-JF)6-)9-&$9-)+,)"7-).J-%-,&-)<br />

+O)"7-)Q$,-)S8)&#Q&PQ#"$,H)"7-).J+S#S$Q$"8F<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

W-O$,$"$+,%<br />

• I)&7+2$ J)(*)6,'K %& 1%0(%>?#& ,4(94& 9%"& $%.#& +"& 2(66#0#"$&<br />

"B)#0(9%?& 1%?B#-G& 9+00#-7+"2("*& $+& 2(66#0#"$& #L7#0()#"$%?&<br />

+B$9+)#-U&<br />

– DL%)7?#;&%&>(""#2&2%$B) D i , ,4(94&9%"&4%1#&2(66#0#"$&1%?B#-&<br />

#1#"&,4#"&%"&#L7#0()#"$&(-&0#7#%$#2&#L%9$?8U&<br />

• 3")"*1"*4K&%&6B"9$(+"&+6&0%"2+)&1%0(%>?#-U&<br />

– DL%)7?#;&%&2%$B) D i , +0&%&7+7B?%$(+"&)#%"<br />

N ( i 1 Di / N)<br />

µ = ⎡⎣ ∑ = ⎤⎦<br />

• L(+6)6*,*"/$1)2M,*&8$7*1"(*6:"*+&K $4#&"+0)%?(T#2&2(-$0(>B$(+"&<br />

60+)& ,4(94& %& -$%$(-$(9& (-& -%)7?#2U& 'B94& %& 2(-$0(>B$(+"& (-&<br />

9+))+"?8&2#"+$#2 p (X | Y ), “$4#&70+>%>(?($8&+6&+B$9+)#&j&*(1#"&<br />

9+"2($(+"O-S&NG`&+0&-+)#$()#-&KB-$&M$O-$SU&V+$#&$4%$&("&$4#&-7#9(%?&<br />

9%-#& +6& $4#& e%B--(%"& O+0& "+0)%?S& 2(-$0(>B$(+"G& M $O- $S& )%8& >#&<br />

,0($$#"&%- N(!,σ 2 ), ,4#0# !$(-&$4#&e%B--(%"&)#%"G&%"2 σ 2 (-&($-&<br />

1%0(%"9#U&<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


F4#&2(-90#$#&=+(--+"&2(-$0(>B$(+";<br />

&&&&&&70+>OA ( S\<br />

70+>%>(?($8&+6&6("2("*&O * &#1#"$-&O4+:&"1S&("&>("&*<br />

O'&'(8/$()8'S&+6&2%$%-#$&O$O1M'4"(:2S&("&%&<br />

*(1#"&?#"*$4&+6&$()#&O#L7+-B0#&$()#SG&(6&$4#&<br />

#1#"$-&+99B0&("2#7#"2#"$?8&%$&%&9+"-$%"$&0%$#&<br />

C *$$ P1+:(4'$*&"'&1*"/QR&<br />

F4("*-&$+&0#)#)>#0;<br />

• :#%" != E [D i] = M i<br />

• k%0(%"9#;& V [D i ] = Mi • cov [D i1 , D i ] = 0 = > &("2#7#"2#"$<br />

2<br />

• $4#&-B)&+6 n =+(--+"32(-$0(>B$#2&<br />

1%0(%>?#-&(-&($-#?6&=+(--+"3<br />

2(-$0(>B$#2&,($4&1%0(%"9#;& n M ∑<br />

3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

17-)\+$%%+,)W$%"J$SP"$+,<br />

5+,,'4"*&8$-.()/$7)")$$ST$5+:&"*&8$*&7*J*7:),$M#+"+&1$<br />

$$$$$$$$$$$$$$$$$ST$3)2M,*&8$U(+2$L+*11+&$7*1"(*6:"*+&<br />

Di<br />

M i<br />

p( Di | M i ) = e<br />

D !<br />

i<br />

−M<br />

i<br />

i= 1 i<br />

i<br />

M → ∞<br />

!-&&&&&&&&&&&&=+(--+"&2(-$0(>B$(+"&<br />

9+"1#0*#-&$+&e%B--(%"&2(-$0(>B$(+"<br />

&VOV&\&C * &l&σ P &\&C * &S<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


5L#K.Q-'<br />

E"$#*#0&9+B"$-&-7#9$0B)&-%)7?#2&60+)&%&9+"-$%"$&%)7?($B2#&)+2#?&,($4&)#%"&V$\&XQ&9+B"$-G&<br />

%"2&6($&,($4&%&7%0%>+?(9&)+2#?U&<br />

3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


5L#K.Q-i<br />

DL%)7?#&+6&%&$,+32()#"-(+"%?&("$#*#0&9+B"$-&-7#9$0B)U&&W+M$X'U"K$5#)&7()&!5E'3'&2%$%&+6&j3<br />

0%8&9?B-$#0&&:'&PZRmUR3PRYRG&,($4&71=&-+B09#&0#*(+"-&-B7#0()7+-#2U&<br />

W+M$I*8#"K&a#-$36($&+6&%&$,+32()#"-(+"%?&>#$%&)+2#?&$+&$4#&6(?$#0#2&2%$%U&<br />

Y+""+2$X'U"K&f#-(2B%?-&O("&B"($-&+6 σ S&+6&$4#&>#-$&6($U&&&<br />

Y+""+2$I*8#"K&F4#&%77?(#2&6(?$#0l&$4#&2%$%&,($4("&$4#&+1%?-&,#0#&#L9?B2#2&60+)&$4#&6($U&<br />

3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


\+$%%+,)`%F)


\+$%%+,)`%F)B$(+"-&O2+$$#2S&+6&)#%" ! = !"#$!%&!'()!*#!+,-.!(/01'2!),3-0,45-,/(3!/6!-.7!3'17!<br />

17'(!'()!8'0,'(97!:;'),7!!"#$%&!"<br />

lEC?<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />


3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

!"#$%&'()*+*$(,-".%"+<br />

17-)"JP-)$K.+J"#,&-)+O)"7-)<br />

$$$$$$$$$$<br />

PC .VQ<br />

& ST$d):11*)&$<br />

O*1"(*6:"*+&<br />

$\[^& $$$$$$$$$)1$&" c$$$$$$$$$$$$<br />

$ $$$$$$VS?9$\ > S;<br />

-("*?#<br />

%1#0%*#-&<br />

+6&p<br />

%1#0%*#-&<br />

+6&P<br />

%1#0%*#-&<br />

+6&&ZX<br />

PQQ&8&1%?B#-&20%,"&60+)&#L7O3LS&6B"9$(+"<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

(#8-%$#,)`%F)*Q#%%$&#Q&<br />

5L#K.Q-'&<br />

&&&&&&&&&&&&&&W)E)@F>#?F=)Z.&)<br />

A+#-&"+$&2#-90(>#&70+>%>(?($8&$4%$&%&$0B#&1%?B#&(-&>#$,##"&qUp&%"2&qUXU<br />

J#&#%%PK-&$4%$&%&e%B--(%"&2(-$0(>B$(+"&%77?(#-&%"2&."+,("*&$4#&2(-$0(>B$(+"&<br />

+6&#00+0-&,#&9%"&)%.#&70+>%>(?(-$(9&-$%$#)#"$-U<br />

*Q#%%$&#Q)I..J+#&7')<br />

!--B)("*&$4#&$0B#&2(-$%"9#&O ? &&$4#"&<br />

O&(-&"+0)%??8&2(-$0(>B$#2&%0+B"2&&O ?& &<br />

,($4&%&-$%"2%02&2#1(%$(+"&+6&QUZU&<br />

f#7#%$("*&)#%-B0#)#"$&,(??&8(#?2&<br />

)%"8&#-$()%$#-&+6&2(-$%"9#$O&&,4(94&<br />

%??&-9%$$#0&%0+B"2&$0B#&O ?R<br />

011:2'$"#'$"#*&8$P7*1")&4'Q$@'$<br />

@)&"$"+$A&+@$)&7$"',,$:1$<br />

#+@$"#'$7)")$@*,,$6'#)J'R<br />

(#8-%$#,)I..J+#&7'<br />

A#2B9#&2(0#9$?8&$4#&70+>%>(?($8&<br />

2(-$0(>B$(+"&+6&O ? &60+)&$4#&2%$%U&<br />

011:2'1$"#'$7)")$)&7$"',,$:1$"#'$<br />

"#*&8$@'$@)&"$"+$A&+@R$H+$<br />

('M'"*"*+&$+U$'FM'(*2'&"R<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


5L#K.Q-'<br />

3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

67#")9+):-)J-#QQ8)9+;<br />

Ed1#&+>-#01#2&)8&-+B09#G&0#2B9#&$4#&2%$%&%"2&6("%??8&*+$&)8&j30%8&<br />

-7#9$0B)&H&&,4%$&2+&E&2+&"+,+B$&$4#&748-(9-&+6&)8&-+B09#<<br />

fB"&m!\5*&+0&3#'(M)[&aB$&,4%$&2+&$4+-#&70+*0%)-&0#%??8&2+<<br />

[$")"7-)9#"#&\]&*T7UEnMT5R7U)IT5U)ZT5RoU95)<br />

5+B"$-<br />

:+2#?<br />

f#-7+"-#<br />

&&&&&D66#9$(1#&!0#%<br />

43&2#$#9$+0&94%""#?-<br />

D3&D"#0*8<br />

r3&)+2#?&7%0%)#$#0-<br />

!--B)#&%&)+2#?&%"2&?++.&6+0&$4#&>#-$&)+2#?&<br />

7%0%)#$#0-&,4(94&&2#-90(>#-&$4#&+>-#01#2&<br />

-7#9$0B)U<br />

4--9)#)\#J#K-"-J)5%"$K#"+J)])!"#"$%"$&%<br />

54%"20%&!5E'3'<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

!"#"$%"$&%<br />

T=U)!"#"$%"$&%)$,9$&#"$,H)"7-)Q+&#"$+,)+O)"7-)9#"#')<br />

)<br />

&&&!1#0%*#;&&&ij]&\&OZ@VS&c ( &&j ( &<br />

&&&:+2#;&?+9%$(+"&+6&$4#&7#%.&("&$4#&4(-$+*0%)l&$4#&1%?B#&<br />

+99B0("*&)+-$&60#IB#"$?8<br />

TiU)!"#"$%"$&%)$,9$&#"$,H)"7-)%&#Q-)+J)#K+P,")+O)%&#""-J'<br />

))):#%"&2#1(%$(+";&isj]&\&OZ@VS&c ( &&bj ( &3ij]b<br />

&&&:#%"&-IB%0#&2#1(%$(+";&' P &\&OZ@VS&c ( &&Oj ( &3ij]S P<br />

&&&&& f++$&:#%"&'IB%0#&2#1(%$(+";&0)-&\&'<br />

&<br />

&&&<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

\#J#K-"-J)5%"$K#"+J%<br />

M-XP$J-K-,"%)+,)!"#"$%"$&%'<br />

• 0,S$#%-9))<br />

])&+,`-JH-)"+)"JP-)`#QP-):$"7)<br />

J-.-#"-9)K-#%PJ-K-,"%<br />

• M+SP%")<br />

f)Q-%%)#OO-&"-9)S8)+P"Q$-J%<br />

• *+,%$%"-,")<br />

f)"JP-)`#QP-)O+J)#)Q#JH-)%#K.Q-)<br />

%$D-)T5L#K.Q-')JK%)#,9)<br />


E6&$4#&487+$4#-(T#2 θ (-&9?+-#&$+&$4#&$0B#&1%?B#G&$4#"&,#&#L7#9$&<br />

%&4(*4&70+>%>(?($8&$+&*#$&2%$%&?(.#&$4%$&,4(94&,#&%9$B%??8&6+B"2U<br />

'+&,#&2#6("#&$4#&)%L()B)&?(.#?(4++2&O:CS&#-$()%$+0O-S&$+&>#&<br />

$4#&7%0%)#$#0&1%?B#O-S&6+0&,4(94&$4#&?(.#?(4++2&(-&)%L()B)U<br />

3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


t"#& 9%"& B-#& $4#& =+(--+"& 2(-$0(>B$(+"& $+& %--#--& $4#& 70+>%>(?($8& +6&<br />

-%)7?("*&%&2%$B)&O * &*(1#"&%&70#2(9$#2&O9+"1+?1#2S&)+2#?&%)7?($B2# C * U<br />

F4B-&$+&%--#--&$4#&IB%?($8&+6&%&6($G&($&(-&"%$B0%?&$+&)%L()(T#&$4#&70+2B9$&<br />

+6&=+(--+"&70+>%>(?($(#-&("&#%94&2%$%&>("G&*R'UG&$+&)%L()(T#&$4#&=+(--+"&<br />

?(.#?(4++2;&<br />

E"&70%9$(9#G&,4%$&(-&+6$#"&)%L()(T#2&(-&$4#&?+*3?(.#?(4++2G<br />

X&\&?+*uU !&,#??3."+,"&-$%$(-$(9&("&j30%8&%-$0+"+)8&,4(94&(-&0#?%$#2&<br />

$+&X&(-&$4#&-+39%??#2&_5%-4&-$%$(-$(9`;&<br />

3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

Z#L$KPK)2$c-Q$7++9'<br />

I%%-%%$,H)"7-)YP#Q$"8)+O)[$"<br />

D<br />

N N i M<br />

N<br />

i<br />

# = ∏# " i = ∏ exp( − M i ) = ∏ p(<br />

Di | M i )<br />

i i D !<br />

i<br />

N<br />

C ≡ 2 ∑[<br />

M − D log M ] $<br />

− 2 L,<br />

i<br />

i<br />

i i i<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

T4+,]U)0%-)+O)"7-)\+$%%+,)2$c-Q$7++9<br />

E"&)+2#?&6($-G&$4#&=+(--+"&?(.#?(4++2&(-&"+$&%-&9+))+"?8&B-#2&%-&($&<br />

-4+B?2&>#U&'+)#&0#%-+"-&,48&("9?B2#;&<br />

• %&4(-$+0(9%?&%1#0-(+"&$+&9+)7B$("*&6%9$+0(%?-l&<br />

• $4#&6%9$&$4#&?(.#?(4++2&9%""+$&>#&B-#2&$+&6($&_>%9.*0+B"2&-B>$0%9$#2`&<br />

-7#9$0%l&<br />

• $4#&6%9$&$4%$&"#*%$(1#&%)7?($B2#-&%0#&"+$&%??+,#2&O"+$&%&>%2&$4("*&<br />

748-(9-&%>4+0-&"#*%$(1#&6?BL#-[Sl<br />

• $4#&6%9$&$4%$&$4#0#&(-&"+&_*++2"#--&+6&6($v&90($#0(+"G&(U#U&$4#0#&(-&"+&#%-8&<br />

,%8&$+&("$#070#$&u )%L &O4+,#1#0G&4UU&$4#&*!1I1&-$%$(-$(9Sl&%"2&<br />

• $4#&6%9$&$4%$&$4#0#&(-&%"&%?$#0"%$(1#&("&$4#&e%B--(%"&?()($;&$4#&% P &<br />

-$%$(-$(9U&<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

% P&) !"#"$%"$&<br />

A#6("($(+";&&&% 2 \&'(& OA ( 3: ( S P @: (<br />

F4#&% 2 &-$%$(-$(9-&(-&K$,$K$D-9&("&$4#&6($$("*&$4#&<br />

2%$%G&1%08("*&$4#&)+2#?&7%0%)#$#0-&B"$(?&$4#&>#-$3<br />

6($&)+2#?&7%0%)#$#0-&%0#&6+B"2&6+0&$4#&)("()B)&<br />

1%?B#&+6&$4#&% 2& -$%$(-(9<br />

A#*0##-3+6360##2+)&\&c]=])4&<br />

V&H&"B)>#0&+6&7%0%)#$#0w&H&"B)>#0&+6&-7#9$0%?&>("-<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

*+,O$9-,&-)2$K$"%<br />

D--#"$(%?&(--B#&\&%6$#0&$4#&>#$-36($&7%0%)#$#0-&%0#&6+B"2&<br />

#-$()%$#&$4#&9+"6(2#"9#&?()($-&6+0&$4#)U&F4#&0#*(+"&+6&<br />

9+"6(2#"9#&(-&*(1#"&>8&O!1"(&ZnmXS;<br />

*&3&2#*0##-&+6&60##2+)<br />

(&3&-(*"(6(9%"9#<br />

% P<br />

)(" &&3&)("()B)&<br />

% P<br />

( \%P<br />

)("& h)O*G(S<br />

)&+&7'M'&71$+&,/$+&$"#'$&:26'($+U$<br />

$$$$$$$$$M)()2'"'(1$*&J+,J'7<br />

$$$$$$$$&+($+&$8++7&'11$+U$U*"<br />

!$H,$O$&#,&-))))))4PKS-J)+O).#J#K-"-J%<br />

))))( ))))=))))))i))))))))G<br />

)<br />

)?Fp@ )=F??))iFG?)))GF>?<br />

)?Fq? )iFB=))CFp=)))pFi><br />

)?Fqq) )pFpG))qFi=)))==FG?)))<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

*#Q&PQ#"$,H*+,O$9-,&-)2$K$"%)<br />

K-#,%))5L.Q+J$,H))"7-)\#J#K-"-J)<br />

!.#&-)])!"#"$%"$&#Q)!PJO#&-<br />

DL%)7?#&+6&%&_,#??3>#4%1#2`&-$%$(-$(9%?&-B06%9#&("&<br />

7%0%)#$#0&-7%9#G&1(#,#2&%-&%&)B?$(32()#"-(+"%?&<br />

7%0%>+?+(2 (χ 2 , top), %"2&%-&%&)B?$(32()#"-(+"%?&<br />

e%B--(%" (exp (- χ 2 / 2) , L, bottom ).<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


(-7#`$+PJ)+O)!"#"$%"$&%)O+J)/,-)\#J#K-"-J<br />

5+)7%0(-+"&+6&F,+&)#$4+2-&("&'4#07%<br />

3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


*+,O$9-,&-)2$K$"%)O+J)1:+)\#J#K-"-J%<br />

f$Y'1"$U*"$M)()2'"'(1<br />

;\9$>\9$b\$4+&"+:(1<br />

5+)7%0(-+"&+6&F,+&)#$4+2-&("&'4#07%<br />

3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


F4#& 1#0-(+"& +6 χ2 2#0(1#2& %>+1#& (-& 2B>>#2& _2%$%& 1%0(%"9#` χ2 , or ,<br />

>#9%B-#&+6&$4#&70#-#"9#&+6&O&("&$4#&2#"+)("%$+0. e#"#0%??8G&$4# χ2 d<br />

-$%$(-$(9&<br />

(-&,0($$#"&%-;&<br />

N<br />

2<br />

σ<br />

,4#0# i 0#70#-#"$-& $4#& OB"."+,"[S& 1%0(%"9#& +6& $4#& =+(--+"& 2(-$0(>B$(+"&<br />

60+)&,4(94&O *$ (-&-%)7?#2U&<br />

χ 2 !"#"$%"$&<br />

3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

2<br />

sN-J%$+,%t)+O)"7- χ 2 !"#"$%"$&<br />

( D − M )<br />

2<br />

i i<br />

χ ≡ ∑ 2<br />

i σ i<br />

A%$%&k%0(%"9# &O *$<br />

:+2#?&k%0(%"9# &C *<br />

e#40#?-<br />

[1 +<br />

2<br />

Di<br />

+ 0.75]<br />

=0()("( &C *$ 60+)&70#1(+B-&>#-$36($<br />

54B0%T+1 >%-#2&+"&12++"#'7&2%$%&A<br />

_=%0#"$`<br />

N<br />

i 1 i<br />

C#%-$&'IB%0#- 1<br />

D ∑<br />

=<br />

N<br />

V+$#&$4%$&-+)#&j30%8&2%$%&%"%?8-(-&0+B$("#-&)%8&#-$()%$# σ i 6+0&8+B&2B0("*&<br />

2%$%&0#2B9$(+"U&E"&=/!&6(?#-G&-B94&#-$()%$#-&%0#&0#9+02#2&("&$4#&!1I1r5MM<br />

9+?B)"U&<br />

,<br />

2<br />

σ i<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??<<br />

χ<br />

2


3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

!"#"$%"$&#Q)3%%P-%<br />

• a(%-<br />

• e++2"#--&+6&^($<br />

• a%9.*0+B"2&'B>$0%9$(+"<br />

• f#>(""("*<br />

• D00+0-<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

!"#"$%"$&#Q)3%%P-%')($#%<br />

• E6&+"#&-%)7?#-&%&?%0*#&"B)>#0&+6&2%$%-#$-&60+)&%&*(1#"&)+2#? %"2&<br />

$4#"&6($-&$4(-&-%)#&)+2#?&$+&$4#-#&2%$%-#$-&O,4(?#&?#$$("* θ 1%08SG&+"#&<br />

,(??&>B(?2&B7&-%)7?("*&2(-$0(>B$(+"-&6+0&#%94&7%0%)#$#0 θ k .<br />

• !"&#-$()%$+0&O'R8., χ2 ) (-&>(%-#2&(6&$4#&)#%"&+6&$4#-#&2(-$0(>B$(+"- (E[θ k])<br />

2(66#0-&60+)&$4#&$0B#&1%?B#- θ k,o.<br />

• F4#&=+(--+"&?(.#?(4++2&(-&%"&B">(%-#2&#-$()%$+0U<br />

• F4# χ2 -$%$(-$(9&4)&&>#&>(%-#2G&2#7#"2("*&B7+"&$4#&94+(9#&+6 σ :<br />

– W-("*&$4#&Sherpa B$(?($8 FAKEIT, ,#&-()B?%$#2&YQQ&2%$%-#$-&60+)&%&9+"-$%"$&<br />

)+2#?&,($4&%)7?($B2#&ZQQ&9+B"$-U&<br />

– J#& $4#"& 6($& #%94& 2%$%-#$& ,($4& %& 9+"-$%"$& )+2#?G& 0#9+02("*& $4#& ("6#00#2&<br />

%)7?($B2#U&<br />

ˆ M θ<br />

!"#"$%"$& ))) Z-#,)IK.Q$"P9e#40#?-<br />

&&&&&&&&&&&&&&&&nnUQY<br />

A%$%&k%0(%"9# &&nnUQP<br />

:+2#?&k%0(%"9# ZQQUpm<br />

_=%0#"$`&&& &&&&&&&&&nnUnp<br />

=0()("( &&nnUnp<br />

5%-4 &&nnUnq<br />

( )<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


!&2#)+"-$0%$(+"&+6&>(%-U&&^(1#&4B"20#2&2%$%-#$-&%0#&-%)7?#2&60+)&%&9+"-$%"$&)+2#?&,($4&%)7?($B2#&ZQQ&%"2&<br />

$4#"& %0#& 6($& ,($4& $4#& -%)#& 9+"-$%"$& %)7?($B2#& )+2#?G& B-("* χ 2 ,($4& 2%$%& 1%0(%"9#U& F4#& )#%"& +6& $4#&<br />

2(-$0(>B$(+"&+6&6($&%)7?($B2#&1%?B#-&(-&"+$&ZQQG&%-&($&,+B?2&>#&(6&$4#&-$%$(-$(9&,#0#&%"&B">(%-#2&#-$()%$+0U&<br />

3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


• Th e χ 2 *++2"#--3+636($&(-&2#0(1#2&>8&9+)7B$("*<br />

α<br />

F4(-&9%"&>#&9+)7B$#2&"B)#0(9%??8&B-("*G&'R8RG&$4#&?("2?8&*1$&+"$('4+22'&7'7[&<br />

• !&IB(9.y"y2(0$8&%770+%94&$+&>B(?2("*&("$B($(+"&%>+B$&4+,&,#??&8+B0&<br />

)+2#?&6($-&$4#&2%$%&(-&$+&B-#&$4#&('7:4'7 χ 2 , i.e.,<br />

– !&_*++2`&6($&4%-<br />

– If $4#&6($&(-&_$++&*++2`&33&,4(94&)#%"-&OZS&$4#&#00+0>%0-&%0#&<br />

$++&?%0*#G&OPS&&&&&&&&(-&&+"&-%)7?#2&60+)&$4#&% P 2<br />

χobs,r ≈ 1.<br />

2<br />

χobs,r → 0<br />

2 χobs<br />

&2(-$0(>B$(+"G&%"2@+0&ORS&$4#&<br />

2%$%&4%1#&>##"&6B2*#2U<br />

F4#&0#2B9#2 χ 2 -4+B?2&"#1#0&>#&B-#2&("&%"8&)%$4#)%$(9%?&9+)7B$%$(+"&<br />

(6&8+B&%0#&B-("*&($G&8+B&%0#&70+>%>?8&2+("*&-+)#$4("*&,0+"*[&<br />

3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

!"#"$%"$&#Q)3%%P-%')


Figure 7: Com p arison of the d istribu tions of 500 sam p led valu es of χ 2 versu s the exp ected d istribu tion for 99<br />

d egrees of freed om . Top : χ 2 with Geh rels variance. Bottom : χ 2 with d ata variance.<br />

3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


• !&$87(9%?&_2%$%-#$`&)%8&9+"$%("&)B?$(7?#&-7#9$0%G&+"#&9+"$%("("*&-+B09#&<br />

%"2& _>%9.*0+B"2`& 9+B"$-G& %"2& +"#& +0& )+0#& +$4#0-& 9+"$%("("*& +"?8&<br />

_>%9.*0+B"2`&9+B"$-U&<br />

– F4#&_>%9.*0+B"2`&)%8&9+"$%("&9+-)(9&%"2&7%0$(9?#&9+"$0(>B$(+"-G&'"4RG&>B$&<br />

,#d??&(*"+0#&$4(-&9+)7?(9%$(+"&%"2&20+7&$4#&IB+$#&)%0.-U&<br />

• E6&7+--(>?#G&+"#&-4+B?2&)+2#?&>%9.*0+B"2&2%$%;&<br />

⇒ '()B?$%"#+B-?8&6($&%&>%9.*0+B"2&)+2#? MB $+&$4#&>%9.*0+B"2&2%$%-#$O-S&Yg &G&<br />

%"2&%&-+B09#&7?B-&>%9.3&*0+B"2&)+2#? MS + MB $+&$4#&0%,&2%$%-#$&OU&<br />

⇒ F4#&>%9.*0+B"2&)+2#?&7%0%)#$#0-&)B-$&4%1#&$4#&-%)#&1%?B#-&("&>+$4&6($-G&<br />

*R'UG&2+&"+$&6($&$4#&>%9.*0+B"2&2%$%&6(0-$G&-#7%0%$#?8U&<br />

2 2<br />

⇒ :%L()(T# LbL LS+ B +0&)("()(T# B S+B<br />

• /+,#1#0G&)%"8&j30%8&%-$0+"+)#0-&9+"$("B#&$+&-B>$0%9$&$4#&>%9.*0+B"2&<br />

2%$%&60+)&$4#&0%,&2%$%;&<br />

. χ + χ<br />

⎡ n<br />

∑<br />

'<br />

j 1 B ⎤<br />

= i, j<br />

Di = Di − β DtD ⎢ ⎥ .<br />

n<br />

⎢ ∑ j = 1 ⎣<br />

β B t<br />

j Bj<br />

⎥⎦<br />

&&(-&$4#&"B)>#0&+6&>%9.*0+B"2&2%$%-#$-G&"&(-&$4#&+>-#01%$(+"&$()#G&%"2& &<br />

(-& $4#& _>%9.-9%?#`& O*(1#"& >8& $4#& a!5w'5!C& 4#%2#0& .#8,+02& 1%?B#& ("& %&<br />

\_I &6(?#SG& $87(9%??8& 2#6("#2& %-& $4#& 0%$(+& +6& 2%$%& #L$0%9$(+"& %0#%& $+& $+$%?&<br />

2#$#9$+0&%0#%U<br />

3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

!"#"$%"$&#Q)3%%P-%')(#&cHJ+P,9)!PS"J#&"$+,<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


Figu re 8: Top: Best- fit of a p ower- law tim es galactic absorp tion m od el to th e source sp ectru m of su p ernova<br />

rem n an t G21.5- 0.9. Bottom : Best- fit of a sep arate p ower- law tim es galactic absorp tion m od el to the<br />

backgrou n d sp ectru m extracted for th e sam e source.<br />

3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

!"#"$%"$&#Q)3%%P-%')(#&cHJ+P,9)!PS"J#&"$+,<br />

• J48&-B>$0%9$&$4#&>%9.*0+B"2#& 2(66(9B?$& $+& -#?#9$& %"& %770+70(%$#& )+2#?& -4%7#& 6+0& $4#&<br />

>%9.*0+B"2U&<br />

– !"%?8-(-&70+9##2-&6%-$#0G&-("9#&>%9.*0+B"2&2%$%-#$-&%0#&"+$&6($U&<br />

– _E$&,+"d$&)%.#&%"8&2(66#0#"9#&$+&$4#&6("%?&0#-B?$-U`<br />

• J48&"+$&-B>$0%9$&$4#&>%9.*0+B"2


3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

!"#"$%"$&#Q)3%%P-%')M-S$,,$,H<br />

• I'6*&&*&8$ 7)")$ *&J)(*)6,/$ ,')71$ "+$ )$ ,+11$ +U$ 1")"*1"*4),$<br />

*&U+(2)"*+&`&<br />

• f#>(""("*& (-& "+$& "#9#--%08& (6& +"#& B-#-& $4#& =+(--+"& ?(.#?(4++2& $+&<br />

)%.#&-$%$(-$(9%?&("6#0#"9#-U&<br />

• /+,#1#0G&$4#&0#>(""("*&+6&2%$%&)%8&>#&"#9#--%08&$+&B-#&χ P &-$%$(-$(9-G&<br />

(6& $4#& "B)>#0& +6& 9+B"$-& ("& %"8& >("& (-& i\ &YU& E"& j30%8& %-$0+"+)8G&<br />

0#>(""("*&O+0&8(+:M*&8S&+6&2%$%&)%8&>#&%99+)7?(-4#2&,($4;&<br />

– grppha, %"&hWiiX3&0+B$("#l&+0<br />

– dmgroup, %&5%0i&A%$%&:+2#?&C(>0%08&0+B$("#U&<br />

t"#&9+))+"&90($#0(+"&(-&$+&-B)&$4#&2%$%&("&%2K%9#"$&>("-&B"$(?&$4#&<br />

-B)&#IB%?-&6(1#&O+0&)+0#SU&<br />

5%1#%$;& %?,%8-& #-$()%$#& $4#& #00+0-& ("& 0#>(""#2& -7#9$0%& B-("*& $4#& '<br />

Di<br />

"#,& 2%$%& & & & & ("& #%94& "#,& >("& O-("9#& $4#-#& 2%$%& %0#& -$(??& =+(--+"3<br />

2(-$0(>B$#2SG&0%$4#0&$4%"&70+7%*%$("*&$4#&#00+0-&("&#%94&+?2&>("U&<br />

⇒ ^+0& #L%)7?#G& (6& $40##& >("-& ,($4& "B)>#0-& +6& 9+B"$- ZG& RG& %"2& Z& %0#&<br />

*0+B7#2& $+& )%.#& +"#& >("& ,($4& Y& 9+B"$-G& +"#& -4+B?2& #-$()%$# V[D ’ = 5]<br />

an d n ot V[D ’ ] = V[D 1 = 1] + V[D 2 = 3] + V [D 3 = 1].&F4#&70+7%*%$#2&#00+0-&<br />

)%8&+1#0#-$()%$#&$4#&$0B#&#00+0-U&<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

!"#"$%"$&#Q)3%%P-%')!8%"-K#"$&)5JJ+J%<br />

• E"& j30%8& %-$0+"+)8G& +"#& B-B%??8& -7#%.-& +6& $,+& $87#-& +6& #00+0-;&<br />

-$%$(-$(9%?&#00+0-G&%"2&-8-$#)%$(9&#00+0-U&<br />

• '8-$#)%$(9& #00+0-& %0#& B"9#0$%("$(#-& ("& ("-$0B)#"$%?& 9%?(>0%$(+"U& ^+0&<br />

("-$%"9#;&<br />

– !--B)#& %& & -7#9$0B)& +>-#01#2& 6+0& $()#& " &,($4& %& $#?#-9+7#& ,($4&<br />

7#06#9$&0#-+?B$(+"&%"2&%"&#66#9$(1#&%0#%&0 * &U&^B0$4#0)+0#G&%--B)#&<br />

$4%$&$4#&B"9#0$%("$8&("&0 * &(- σ A ,i .<br />

– V#*?#9$("*& 2%$%& -%)7?("*G& ("& >("& *G& $4#& #L7#9$#2& "B)>#0& +6&<br />

9+B"$-&(- D i = D γ ,i(ΔE )tA i.<br />

– J#&#-$()%$#&$4#&B"9#0$%("$8&("&O * $%-<br />

σD i = D γ ,i (ΔE )tσ A ,I = D γ ,i (ΔE )tf i A i = f i D i<br />

• F4#&-8-$#)%$(9&#00+0&U * O *$ l&("&=/!&6(?#-G&$4#&IB%"$($8&U *$ (-&0#9+02#2&("&<br />

$4#&'z'{Dff&9+?B)"U&<br />

• '8-$#)%$(9&#00+0-&%0#&%22#2&("&IB%20%$B0#&,($4&-$%$(-$(9%?&#00+0-l&6+0&<br />

2<br />

("-$%"9#G&(6&+"#&B-#-&&&&&&&&$+&%--#--&$4#&IB%?($8&+6&6($G&$4#"&&<br />

χd<br />

σ i = Di +<br />

fiDi 2<br />

• F+&B-#&("6+0)%$(+"&%>+B$&-8-$#)%$(9&#00+0-&("&%&=+(--+"&?(.#?(4++2&<br />

6($G&+"#&)B-$&("9+07+0%$#&$4(-&("6+0)%$(+"&("$+&$4#&)+2#?G&%-&<br />

+77+-#2&$+&-()7?8&%2KB-$("*&$4#&#-$()%$#2&#00+0&6+0&#%94&2%$B)U&<br />

( ) .<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

!PKK#J8<br />

• :+$(1%$(+";&,48&2+&,#&"##2&-$%$(-$(9-<<br />

• =0+>%>(?($(#-@A(-$0(>B$(+"-<br />

• =+(--+"&C(.#?(4++2<br />

• =%0%)#$#0&D-$()%$(+"<br />

• '$%$(-$(9%?&E--B#-<br />

• '$%$(-$(9%?&F#-$-&H&-$(??&$+&9+)#UUUU<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

*+,&QP%$+,%<br />

!"#"$%"$&%)$%)"7-)K#$,)"++Q)O+J)#,8)#%"J+,+K-J):7+)<br />

,--9)"+)9+)9#"#)#,#Q8%$%)#,9),--9)"+)9-&$9-)#S+P")<br />

"7-).78%$&%).J-%-,"-9)$,)"7-)+S%-J`#"$+,%F<br />

f#6#0#"9#-;<br />

=#$#0&^0##)%"d-&C#9$B0#-&60+)&$4#&=%-$&j30%8&!-$0+"+)8&'94++?;<br />

4$$7;@@L0%8-94++?U*-69U"%-%U*+1@2+9-@L0%8-94++?3PQQR@$%?.-U4$)?<br />

_=0%9$(9%?&'$%$(-$(9-&6+0&!-$0+"+)#0-`G&J%??&M&N#".("-G&PQQR<br />

&5%)>0(2*#&W"(1#0-($8&=0#--<br />

D%2(#&#$&%?&ZnmXG&_'$%$(-$(9%?&:#$4+2-&("&DL7#0()#"$%?&=48-(9-`<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


e#"#0%?&-$%$(-$(9-;<br />

3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

!-Q-&"-9)M-O-J-,&-%<br />

a%>BG&eU&NUG&^#(*#?-+"G&DU&AU&ZnnXG&!-$0+-$%$(-$(9-&OC+"2+";&54%7)%"&M&/%??S<br />

D%2(#G&JU&FUG&A0(K%02G&AUG&N%)#-G&^U&DUGf++-G&:UG&M&'%2+B?#$G&aU&ZnmZG&'$%$(-$(9%?&:#$4+2-&("&DL7#0()#"$%?&=48-(9-&<br />

O!)-$#02%);&V+0$43/+??%"2S<br />

=0#--G& JU& /UG& F#B.+?-.8G& 'U& !UG& k#$$#0?("*G& JU& FUGM& ^?%""#08G& aU& =U& ZnnPG& VB)#0(9%?& f#9(7#-& O5%)>0(2*#;&<br />

5%)>0(2*#&W"(1U&=0#--S<br />

E"$0+2B9$(+"&$+&a%8#-(%"&'$%$(-$(9-;<br />

C+0#2+G& FU& NU& ZnnPG& ("& '$%$(-$(9%?& 54%??#"*#-& ("& :+2#0"& !-$0+"+)8G#2U& DU& ^#(*#?-+"& M& eU& a%>B& OV#,& z+0.;&<br />

'70("*#03k#0?%*SG&PmY<br />

:+2(6(#2&-%"2&% P &'$%$(-$(9-;<br />

5%-4G&JU&ZnmnG&!7N&PPqG&nRn|($#)&54B0%T+1G&DUG&#$&%?U&ZnnXG&!7N&pmZG&XmR<br />

e#40#?-G&VU&ZnqXG&!7N&RQRG&RRX<br />

&w#%0"-G&wUG&=0()("(G&^UG&M&!?#L%"2#0G&AU&ZnnYG&("&!-$0+"+)(9%?&A%$%&!"%?8-(-&'+6$,%0#&%"2&'8-$#)-&EkG#2-U&fU&!U&<br />

'4%,G&/U&DU&=%8"#G&M&NU&NU&DU&/%8#-&O'%"&^0%"9(-9+;&!'=SG&RRZ<br />

E--B#-&("&^($$("*;<br />

^0##)%"G&=U&DUG&#$&%?U&ZnnnG&!7N&YPpG&mYR&O%"2&0#6#0#"9#-&$4#0#("S<br />

3#'(M)&%"2&-3Lj5;<br />

^0##)%"G&=U&DUG&A+#G&'UG&M&'(#)(*("+,-.%G&!U&PQQZG&%-$0+374@QZQqpPX&<br />

4$$7;@@%-9U4%01%02U#2B@9(%+@2+,"?+%2@2+9@-4#07%{4$)?{)%"B%?@("2#LU4$)?<br />

!0"%B2G&wU&!U&ZnnXG&("&!-$0+"+)(9%?&A%$%&!"%?8-(-&'+6$,%0#&%"2&'8-$#)-&kG&#2-U&eU&/U&N%9+>8&M&NU&a%0"#-&O'%"&<br />

^0%"9(-9+;&!'=SG&Zm<br />

4$$7;@@4#%-%09U*-69U"%-%U*+1@2+9-@L%"%2B@L-7#9@)%"B%?@)%"B%?U4$)?<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??


3")"*1"*419$<br />

0&'")$3*'2*8*&+@1A)<br />

\J+.-J"$-%)+O)W$%"J$SP"$+,%<br />

F4#& >#*(""("*& j30%8& %-$0+"+)#0& +"?8& "##2-& $+& >#& 6%)(?(%0& ,($4& 6+B0& 70+7#0$(#-& +6&<br />

2(-$0(>B$(+"-;&$4#&)#%"G&)+2#G&1%0(%"9#G&%"2&-$%"2%02&2#1(%$(+"G&+0&_#00+0U`<br />

• :#%";& ! = E[X ] = !dX X p(X)<br />

• A/)7B m ax[p (X)]<br />

2 2<br />

• k%0(%"9#; V[ X ] = E[( X − µ ) ] = ∫ dX ( X − µ ) p( X )<br />

• ;00/0B σ X = V[ X ]<br />

V+$#&$4%$&(6&$4#&2(-$0(>B$(+"&(-&e%B--(%"G&$4#"&}&(-&("2##2&$4#&e%B--(%"&$\$O4#"9#&$4#&<br />

"+$%$(+"SU<br />

&<br />

E6&$,+&0%"2+)&1%0(%>?#-&%0#&$+&>#&K+("$?8&9+"-(2#0#2G&$4#"&$4#&-%)7?("*&2(-$0(>B$(+"&(-&<br />

$,+32()#"-(+"%?G&,($4&-4%7#&?+9%??8&2#-90(>#2&>8&$4# 4+J)(*)&4'$2)"(*F;&<br />

⎛ V[ X ] cov[ X , X ]<br />

1 1 2 ⎞<br />

⎜ ⎟<br />

cov[ X , X ] V[ X ]<br />

⎝ 1 2 2 ⎠<br />

F4#&0#?%$#2 4+((',)"*+&$4+'UU*4*'&" is<br />

,4#0#<br />

F4#&9+00#?%$(+"&9+#66(9(#"$&9%"&0%"*#&60+)&3Z&$+&ZU&<br />

cov[ X , X ] = E[( X − µ )( X − µ )]<br />

1 2 1 X1 2 X2<br />

= E[ X X ] − E[ X ] E[ X ]<br />

cov[ X , X ]<br />

1 2<br />

corr[ X 1, X 2]<br />

=<br />

.<br />

σ X σ 1 X2<br />

1 2 1 2<br />

! "# $%&"'(&)"*+&),$-.()/$01"(+&+2/$34#++,$$$$$$$$$$$$<br />

$$$$$$$$5)26(*78'9$0:8:1"$;??

Hooray! Your file is uploaded and ready to be published.

Saved successfully!

Ooh no, something went wrong!