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

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Figu re 7 : Com p ar ison of the d is t r ibu t ions of 500 sam p led valu es of χ2 versu s the exp ected d is t r ibu t ion for 99 d egrees of freed om . Top : χ2 with Geh rels var iance. Bot tom : χ2 with d ata var iance.

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2 2B S+B .χ + χ

1 ,'

1.

j j

nj i j

i i D D nj B B

BD D t

β=

=

= −

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Figu re 8 : Top: Bes t - fit of a p ower- law t im es galact ic absorp t ion m od el to th e sou rce sp ect ru m of su p ernova rem nan t G21.5- 0 .9 . Bottom : Bes t - fit of a sep arate p ower- law t im es galact ic absorp t ion m od el to the backgrou nd sp ect ru m ext racted for th e sam e sou rce.

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'iD

11 1

2

1

2

',

1

[ { ,..., )] cov( , )

[ ]

[ ] [ ] [ ] ) .j j

m m

m i ji j i j

m

ii i

nD D

i i i jj B B

f fV f X X X X

f V X

tV D V D V B

t

µ µ

µ

β

β

= =

=

=

∂ ∂≈

∂ ∂

∂≈ ∂

⇒ ≈ +

∑∑

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2dχ

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[ ]X V Xσ =

2 2[ ] [( ) ] ( ) ( )V X E X dX X p Xµ µ= − = −∫

1 1 2

1 2 2

[ ] cov[ , ]cov[ , ] [ ]

V X X XX X V X

1 21 2 1 2

1 2 1 2

cov[ , ] [( )( )]

[ ] [ ] [ ]X XX X E X X

E X X E X E X

µ µ= − −

= −

1 2

1 21 2

cov[ , ]corr[ , ] .X X

X XX Xσ σ

=