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