STAT 134 Chapter 7: unbiased sample variance
Proof
Thm :SE 't En
,lxi -x)
'
is an unbiased estimator of 02
Pf :Note that ECZXI )=ZECXD and ECCX )= CECX )
Also by HWI ,varcx )=Ecxz )-EC
,
,x:
EH
')=o2tn2
and var CI )=ECIY -ECIT =ELI )#tu
'
we show that Eloy :S
'
ElZlxi -IYKEIZCXI '
-Zxixtx
'))
=ELECXI '
-Ezxi It EIY since Iis constant
=ELZXI
'
-zxzxitnx
')
=Eczxitnxz )
=ELExi2 )-ELNE
)
=ZECXIY -NEH
')
=ZloIn4 -nlhetri )
=no 't nn2 -02 -nn2
=(n-1) 02
Hence FIE (Ecxi -IM =P
'
