‫ﭘﺎﺳﺦ ﺗﻤﺮﯾﻦ ‪) ٩‬ﻓﺼﻞ ھﺸﺘﻢ ﮐﺘﺎب درﺳﯽ(‬
‫‪-1‬‬
‫] ) – ) ( ([‬
‫] ) – ]) ( [ ‪[ ( )] +‬‬
‫– ) ( ([ =‬
‫[ ( ‪[ ( )]) +‬‬
‫– ) ( ([ =‬
‫) – ]) (‬
‫]) – ]) ( [ ()]) ( [‬
‫] ) – ]) ( [ ([ ‪[ ( )]) ] +‬‬
‫) – ]) ( [ (])]) ( [‬
‫ﭼﻮن ) ( ‪،‬‬
‫اﺳﺖ‪ ،‬دارﯾﻢ‪:‬‬
‫‪=0‬‬
‫ﭘﺲ ﻫﺮ ﺿﺮﯾﺒﯽ از آن ﻧﯿﺰ ﺻﻔﺮ ﻣﯽﺷﻮد‪.‬‬
‫– ]) ( [‬
‫– ) ( ([ =‬
‫– ) ( ([ ‪+2‬‬
‫) – ]) ( [ ( ‪[ ( )]) ] +‬‬
‫)) ( (‬
‫– ) ( (‪+‬‬
‫– ) ( ([ =‬
‫‪( ( )) +‬‬
‫=‬
-2
‫ را ﺑﺮاي ﭘﺎراﻣﺘﺮ‬1‫( ﺗﺨﻤﯿﻦ ﺑﯿﺸﺘﺮﯾﻦ درﺳﺖﻧﻤﺎﯾﯽ‬a
.‫ﺑﯿﺎﺑﯿﺪ‬
( ; ) =
1
3
0
∀, −
≤
≤2
. .
=
1
3
(max( ) ≤ 2 ) (min( ) ≥ − )
=
1
3
1
≥ max( max( ) , – min( ))
2
= argmax
1
( ; ) = max( max( ) , – min( ))
2
(b
( | ) ( )
( )
= argmax ( | ) = argmax
( | )=
=
1
3
( | ) ( )
(max( ) ≤ 2 ) (min( ) ≥ − )
1
= max( max( ) , − min( ) , ) = max(
2
1
MLE (Maximum Likelihood Estimation)
2
, )
( ≥
)
-3
(a
1
∀, 0≤
( ; , ) =
∑
0
= argmax
, ̂
∀,
≤ ,
=1
≥ 0, = 2
. .
( ; , )
,
=1 ⇒
=2 ⇒
( ; )=
1
(max( ) ≤ ) (min( ) ≥ 0)
= argmax
( ; )
= max( )
( ; )=
= argmax
∑
(min( ) ≥ 0)
( ; )
= argmax ln( ( ; ))
ln( ( ; ) =
ln( ( ; )
=
∑
‫ﺧﻮاﻫﺪ ﺷﺪ و‬
=
ln −
=
−
=0
1
, ̂
= ( , 1) ‫ آنﮔﺎه‬،‫ﺷﻮد‬
.‫ﻣﯽﺷﻮد‬
;
;
;
, ̂
1
(min( ) ≥ 0)
max( )
1
(min( ) ≥ 0)
=
=
≥
;
‫ﺣﺎل اﮔﺮ‬
= ( , 2) ‫در ﻏﯿﺮ اﯾﻦ ﺻﻮرت‬
‫‪ (b‬ﺑﺎ اﺳﺘﻔﺎده از رواﺑﻂ ﺑﺪﺳﺖ آﻣﺪه دارﯾﻢ‪:‬‬
‫)‪= ( , 2‬‬
‫‪= 6‬‬
‫‪max( ) = 11‬‬
‫‪1 + 1 + 1 + 2 + 2 + 11‬‬
‫=‬
‫‪=3‬‬
‫‪6‬‬
‫‪= max( ) = 11‬‬
‫‪1 1‬‬
‫= =‬
‫‪3‬‬
‫‪min( ) = 1‬‬
‫‪1‬‬
‫;‬
‫=‬
‫‪11‬‬
‫‪1‬‬
‫;‬
‫=‬
‫‪3‬‬
‫;‬
‫<‬
‫;‬
‫⇒‬
‫̂ ‪,‬‬