jan 2010 s2

8/12/2019 jan 2010 S2

1/24

!"#$%&'()* ,*' -&./

0'#$ 1'#2'()* ,*' -&./

3,($' 4&%5%#.6*7

3%8,('

9'&5('

:-;

!"#$ &'(#

9#&2%2#5'

:-;

93;)( ?@; )*2A38213@8 B1)A4 38+,9()( C31D >9)413@8 E2E);4

Q#5S'$#5%L#. T-($,.#' 6H%&@ -( :%.

U(''&7

/28(3(21)4 A2< 94) 28< +2,+9,21@; 2,,@C)( :< 1D) ;)F9,213@84 @? 1D) G@381/@98+3, ?@; H92,3?3+213@84I /2,+9,21@;4 A941 8@1 D27) 1D) ?2+3,31< ?@; 4-='V W(%5' /-,( L'&5(' &,$>'(V L#&2%2#5' &,$>'(V /-,( *,($'V %&%5%#.* #&2 *%8,(';9S'L@ 5S#5 /-, S#=' 5S' L-(('L5 X,'*5%-& I#I'(;Y&*W'( Y11 5S' X,'*5%-&*;Z-, $,*5 W(%5' /-,( #&*W'( 5- '#LS X,'*5%-& %& 5S' *I#L' K-..-W%&8 5S' X,'*5%-&;[#.,'* K(-$ 5S' *5#5%*5%L#. 5 #>.'* *S-,.2 >' X,-5'2 %& K,..; \S'& # L#.L,.#5-( %* ,*'2V 5S' #&*W'( *S-,.2

>' 8%='& 5- #& #II(-I(%#5' 2'8('' -K #LL,(#L/;

B8?@;A213@8 ?@; /28(3(21)4

Y >--@.'5 ]Q#5S'$#5%L#. T-($,.#' #&2 35#5%*5 %L#. 0#>.'*) %* I(-=%2'2;T,.. $#(@* $#/ >' ->5#%&'2 K-( #&*W'(* 5- Y11 X,'*5%-&*;0S' $#(@* K-( %&2%=%2,#. X,'*5%-&* #&2 5S' I#(5* -K X,'*5%-&* #(' *S-W& %& (-,&2 >(#L@'5*R ';8; K5L;0S'(' #(' G X,'*5%-&* %& 5S%* X,'*5%-& I#I'(; 0S' 5-5#. $#(@ K-( 5S%* I#I'( %* GE;0S'(' #(' BD I#8'* %& 5S%* X,'*5%-& I#I'(; Y&/ >.#&@ I#8'* #(' %&2%L#5'2;

6(73+) 1@ /28(3(21)4Z-, $,*5 '&*,(' 5S#5 /-,( #&*W'(* 5- I#(5* -K X,'*5%-&* #(' L.'#(./ .#>'..'2;Z-, *S-,.2 *S-W *,KK%L%'&5 W-(@%&8 5- $#@' /-,( $'5S-2* L.'#( 5- 5S' !"#$%&'(;Y&*W'(* W%5S-,5 W-(@%&8 $#/ &-5 8#%& K,.. L('2%5;

H#I'( J'K'('&L'

! ! " # % &

0S%* I,>.%L#5%-& $#/ >' ('I(-2,L'2 -&./ %& #LL-(2#&L' W%5S!2'"L'. 1%$%5'2 L-I/(%8S5 I-.%L/;

^BOAO !2'"L'. 1%$%5'2;

H(%&5'()* 1-8; :-;

:CEGABY\_EO`JFF_DÈGEGO EÈÈ`C`C

!"#$%&'()&'*!

8/12/2019 jan 2010 S2

2/24

1'#='

>.#&@

B

!"#$%&'()''*!

&I Y $#&,K#L5,('( *,II.%'* a[a I.#/'(* 5- ('5#%.'(* %& >#5LS'* -K BO; 45 S#* Eb -K 5S' I.#/'(*

('5,(&'2 >'L#,*' 5S'/ #(' K#,.5/;

6#7 \(%5' 2-W& # *,%5#>.' $-2'. K-( 5S' 2%*5(%>,5%-& -K 5S' &,$>'( -K K#,.5/ a[a I.#/'(*

%& # >#5LS;

K5L

T%&2 5S' I(->#>%.%5/ 5S#5 # >#5LS L-&5#%&*

6>7 &- K#,.5/ a[a I.#/'(*V

K5L

6L7 $-(' 5S#& D K#,.5/ a[a I.#/'(*;

K5L

627 T%&2 5S' $'#& #&2 =#(%#&L' -K 5S' &,$>'( -K K#,.5/ a[a I.#/'(* %& # >#5LS;

K5L

ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc







cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
















X is distributed not P

8/12/2019 jan 2010 S2

3/24

1'#='

>.#&@

C

M9;8 @7);!"#$%&'()#'*!

H9)413@8 & +@81389)(


































ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc H&

KM@12, " A2;N4L

8/12/2019 jan 2010 S2

4/24

1'#='

>.#&@

D

!"#$%&'()*'*!

5I Y L-&5%&,-,* (#&2-$ =#(%#>.'!S#* L,$,.#5%=' 2%*5(%>,5%-& K,&L5%-&

"dP B

P B"D

"eD

6#7 T%&2 H6!d O7;

K5L

6>7 T%&2 5S' I(->#>%.%5/ 2'&*%5/ K,&L5%-& K6"7 -K!;

KOL

6L7 \(%5' 2-W& 5S' $' -K 5S' 2%*5(%>,5%-& -K!;

K&L

627 T%&2 5S' $'#& #&2 5S' =#(%#&L' -K!;

KOL

6'7 \(%5' 2-W& 5S' =#.,' -K H6! f A7;

K&L






















F ( )

,

,

,

x x

=+

0

2

6

1

X~ not P~

say youre integrating fxdx

continouos uniform not random

8/12/2019 jan 2010 S2

5/24

1'#='

>.#&@

E

M9;8 @7);!"#$%&'()$'*!

H9)413@8 5 +@81389)(




































8/12/2019 jan 2010 S2

6/24

1'#='

>.#&@

F

!"#$%&'()+'*!

H9)413@8 5 +@81389)(




































8/12/2019 jan 2010 S2

7/24

1'#='

>.#&@

G

M9;8 @7);!"#$%&'()%'*!

H9)413@8 5 +@81389)(


































ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc H5

KM@12, &% A2;N4L

8/12/2019 jan 2010 S2

8/24

1'#='

>.#&@

_

!"#$%&'(),'*!

OI Y (->-5 %* I(-8(#$$'2 5- >,%.2 L#(* -& # I(-2,L5%-& .%&'; 0S' (->-5 >('#@* 2-W& #5

(#&2-$ #5 # (#5' -K -&L' '='(/ BO S-,(*;

6#7 T%&2 5S' I(->#>%.%5/ 5S#5 %5 W%.. W-(@ L-&5%&,-,*./ K-( E S-,(* W%5S-,5 # >('#@2-W&;

KOL

T%&2 5S' I(->#>%.%5/ 5S#5V %& #& _ S-,( I'(%-2V

6>7 5S' (->-5 W%.. >('#@ 2-W& #5 .'#*5 -&L'V

KOL

6L7 5S'(' #(' '"#L5./ B >('#@2-W&*;

K5L

4& # I#(5%L,.#( _ S-,( I'(%-2V 5S' (->-5 >(-@' 2-W& 5W%L';

627 \(%5' 2-W& 5S' I(->#>%.%5/ 5S#5 5S' (->-5 W%.. >('#@ 2-W& %& 5S' K-..-W%&8 _ S-,(

I'(%-2; U%=' # ('#*-& K-( /-,( #&*W'(;

K5L
























says p breaks down once

not twice

8/12/2019 jan 2010 S2

9/24

1'#='

>.#&@

M

M9;8 @7);!"#$%&'()-'*!

H9)413@8 O +@81389)(




































8/12/2019 jan 2010 S2

10/24

1'#='

>.#&@

AO

!"#$%&'()&)'*!

H9)413@8 O +@81389)(




































8/12/2019 jan 2010 S2

11/24

1'#='

>.#&@

AA

M9;8 @7);!"#$%&'()&&'*!

H9)413@8 O +@81389)(


































ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc HO

KM@12, &% A2;N4L

8/12/2019 jan 2010 S2

12/24

1'#='

>.#&@

AB

!"#$%&'()&''*!

#I 0S' L-&5%&,-,* (#&2-$ =#(%#>.'!S#* I(->#>%.%5/ 2'&*%5/ K,&L5%-& K6"7 8%='& >/

O d"C

C d"D

-5S'(W%*'

WS'(' $%* # L-&*5#&5;

6#7 3S-W 5S#5 $ A

M

K#L

6>7 T%&2 5S' L,$,.#5%=' 2%*5(%>,5%-& K,&L5%-& T6"7;

K!L

6L7 T%&2 5S' $'#& -K!;

KOL

627 3S-W 5S#5 5S' $'2%#& -K!.%'* >'5W''&"fB;F #&2"fB;G

K#L























f ( )

( )

x

k x x

k=

+

2 2 2

3

0

if 2 pdfs are given add them together

8/12/2019 jan 2010 S2

13/24

1'#='

>.#&@

AC

M9;8 @7);!"#$%&'()'*!

H9)413@8 # +@81389)(




































8/12/2019 jan 2010 S2

14/24

1'#='

>.#&@

AD

!"#$%&'()&*'*!

H9)413@8 # +@81389)(




































8/12/2019 jan 2010 S2

15/24

1'#='

>.#&@

AE

M9;8 @7);!"#$%&'()&$'*!

H9)413@8 # +@81389)(




































8/12/2019 jan 2010 S2

16/24

1'#='

>.#&@

AF

!"#$%&'()&+'*!

H9)413@8 # +@81389)(




































8/12/2019 jan 2010 S2

17/24

1'#='

>.#&@

AG

M9;8 @7);!"#$%&'()&%'*!

H9)413@8 # +@81389)(


































ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc H#

KM@12, &P A2;N4L

8/12/2019 jan 2010 S2

18/24

1'#='

>.#&@

A_

!"#$%&'()&,'*!

QI Y L#Kg *'(='* >('#@K#*5 '='(/ $-(&%&8; 9,*5-$'(* #((%=' K-( >('#@K#*5 #5 (#&2-$ #5 # (#5'

-K A '='(/ F $%&,5'*;

T%&2 5S' I(->#>%.%5/ 5S#5

6#7 K'W'( 5S#& M L,*5-$'(* #((%=' K-( >('#@K#*5 -& # Q-&2#/ $-(&%&8 >'5W''& AO #$ #&2

AA #$;

KOL

0S' L#Kg *'(='* >('#@K#*5 '='(/ 2#/ >'5W''& _ #$ #&2 AB &--&;

6>7 h*%&8 # *,%5#>.' #II(-"%$#5%-&V '*5%$#5' 5S' I(->#>%.%5/ 5S#5 $-(' 5S#& EO L,*5-$'(*

#((%=' K-( >('#@K#*5 &'"5 0,'*2#/;

K!L



























8/12/2019 jan 2010 S2

19/24

1'#='

>.#&@

AM

M9;8 @7);!"#$%&'()&-'*!

H9)413@8 Q +@81389)(


































ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc HQ

KM@12, R A2;N4L

8/12/2019 jan 2010 S2

20/24

1'#='

>.#&@

BO

!"#$%&'()')'*!

!I 6#7 a'K%&' 5S' L(%5%L#. ('8%-& -K # 5'*5 *5#5%*5%L;

K5L

Y 2%*L('5' (#&2-$ =#(%#>.'!S#* # ?%&-$%#. 2%*5(%>,5%-& ?6COV%7; Y *%&8.' ->*'(=#5%-& %*

,*'2 5- 5'*5 iOR%f O;C #8#%&*5 i

AR% O;C

6>7 h*%&8 # Ab .'='. -K *%8&%K%L#&L' K%&2 5S' L(%5%L#. ('8%-& -K 5S%* 5'*5; Z-, *S-,.2 *5#5'

5S' I(->#>%.%5/ -K ('j'L5%-& %& '#LS 5#%. WS%LS *S-,.2 >' #* L.-*' #* I-**%>.' 5- O;OOE

KQL

6L7 \(%5' 2-W& 5S' #L5,#. *%8&%K%L#&L' .'='. -K 5S' 5'*5;

K&L

0S' =#.,' -K 5S' ->*'(=#5%-& W#* K-,&2 5- >' AE;

627 9-$$'&5 -& 5S%* K%&2%&8 %& .%8S5 -K /-,( L(%5%L#. ('8%-&;

K5L








cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
















Text

8/12/2019 jan 2010 S2

21/24

1'#='

>.#&@

BA

M9;8 @7);!"#$%&'()'&'*!

H9)413@8 ! +@81389)(


































ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc H!

KM@12, &% A2;N4L

8/12/2019 jan 2010 S2

22/24

8/12/2019 jan 2010 S2

23/24

1'#='

>.#&@

BC

M9;8 @7);!"#$%&'()'#'*!

H9)413@8 P +@81389)(




































8/12/2019 jan 2010 S2

24/24

1'#='

>.#&@

BD

!"#$%&'()'*'*!

H9)413@8 P +@81389)(

































MSM6T USV W6W'VX PQ =6VY0

'Z[

HP

KM@12, && A2;N4L

jan 2010 s2

Documents