HW S solutions
Q1The likelihood of a response y given Xi
is
119112501 if You
Pry 2 2 0 if Yo 2i
Pry KI X O it Yik
I pry b x g tookkl
If freeYish
This the likelihood of dataset Cti Yo i n
is
Lok II It freeftsonInstead of maximizing L we maximize log L
log LOK É Ig log free
which is equivalent to minimizing
log 401 ÉÉ Ig log free
Q2We have Eo I É for ZEIR
for some b so
For a given Example hi Yi we have
E Yo t.to
So the likelihood of yo given to is
E1 t.to b
By independence of samples the likelihood
of the dataset is
LIO IT Emitter
We have
by Lio É Yogi lograbMaximizing log Lfo Is equivalent to minimizing
log Lrt É Mi total
as b is a constant
X n NIO IaQ3 yo Xt otter w ECE o
Var real 02
a
RCFal IE 119 Farrill
EellX te tell
I Iltrot e tell
E trot OITtacxtro OSE HeliXIE
MEN EÉ ÉÉIÉI
ENTREAIT t O Varied
Ex 10 0 text E o t Varre
0 07 Er I FEEL Varro
Tariancematrix0 07 I rot Ol Varied
110 0 11 t o
b Let e IRdbe such that I X
jthcomponent
Let y f E fi of the vector Xi
The Solution to least squares linear regression
is
E LEET tyWe have
Y X O TE NOTE
Let's write E in terms of Ot and E
XX Xt Kotte
txt xtxottftefateat texter Xt e
soEG Ott o as EntenteE
gEnter'xt e
ECXtxfxt.co
So Q is unbiased since X anole
are independent
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