hw_answers8 - E c o n 3 2 1 A n s w e rs H o m e w o rk 0 8...

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Unformatted text preview: E c o n 3 2 1 A n s w e rs H o m e w o rk 0 8 M a rtin S c h m id t S u m m e r 2 0 1 0 R e v ie w E x e r c is e 1 6 .4 ( m e a n t to illu s tr a te th e la w o f la rg e n u m b e r s ) (a ) 6 0 ro lls a re b e tte r th a n 6 0 0 ro lls . W ith m o re ro lls , th e p e rc e n ta g e o f a c e s w ill b e c lo s e r to 1 6 .7 % ( a s th e p ro b a b ility o f h a v in g a n a c e in 1 ro ll o f a fa ir d ie is 1 / 6 ) . O n e w o u ld lik e th e p e rc e n ta g e o f a c e s to b e fa r fro m 1 6 .7 % s o th a t it c a n b e h ig h e r th a n 2 0 % . C h a n c e e rro r in th e p e rc e n ta g e s d e c re a s e s a s th e n u m b e r o f r o lls in c re a s e s . T h e la w o f la rg e n u m b e rs te lls o n e th a t, in th e in fin ite lim it, th e p e rc e n ta g e o f a c e s w ill b e a s c lo s e to 1 6 .7 % a s o n e d e s ire s . i.e . th e c h a n c e e rro r is v irtu a lly z e ro . (b ) 6 0 0 ro lls a re b e tte r th a n 6 0 ro lls . T h e re a s o n in g is a n a lo g o u s to (a ) , b u t n o w o n e c a n b e s u r e th a t, in th e in fin ite lim it, o n e w ill a lw a y s w in . (c ) 6 0 0 ro lls a re b e tte r th a n 6 0 ro lls . E x a c tly th e s a m e re a s o n in g a s in (b ). (d ) 6 0 ro lls a re b e tte r th a n 6 0 0 ro lls . A s o n e ro ll s m o r e a n d m o re , th e re a re m o r e a n d m o r e p o s s ib ilitie s , n o p a r tic u la r o n e c a n b e v e ry lik e ly . E v e n if a n in c re a s e in ro lls b rin g s th e p e rc e n ta g e o f a c e s c lo s e r a n d c lo s e r to 1 6 .7 % , th e p e r c e n ta g e o f a c e s w ill b e a b it h ig h e r o r lo w e r a s lo n g a s th e n u m b e r ro lls is fin ite . F o r e x a m p le , if o n e m a k e s 6 0 0 0 ro lls o n e c a n g e t 1 0 0 0 a c e s , o r 1 0 0 1 , o r 1 0 0 2 . T h e re a re lo ts o f p o s s ib ilitie s , e a c h o n e in d iv id u a lly h a s a s m a ll c h a n c e . A n d o n ly S tr ic tly s p e a k in g th e la w o f la r g e n u m b e r s ta lk s o n ly a b o u t th e in fin ite li m it. T h e n o r m a l a p p r o x im a tio n o f p r o b a b ility h is to g r a m s c a n s a y s o m e th in g a b o u t a fin ite n u m b e r o f r o lls . O n e c a n a p p ly th e n o r m a l a p p r o x im a tio n o f p r o b a b ility h is to g r a m s to th e r a n d o m v a r ia b le X ’ n = (X 1 + .. + X n ) / n a n d th e n c o n fir m th e a b o v e s ta te m e n ts . R e v ie w E x e r c is e 1 6 .5 ( m e a n t to illu s tr a te th e la w o f la rg e n u m b e r s ) F a ls e . If th e n u m b e r o f h e a d s is 5 0 , th e p e rc e n ta g e o f h e a d s w ill b e 5 0 % . O th e rw is e th e p e rc e n ta g e w ill b e a b it b ig g e r o r a b it s m a lle r . I t is c le a r th a t a n o u tc o m e w ith e x a c tly 5 0 c o i n to s s e s is n o t v e ry lik e ly (it h a s a p r o b a b ility a ro u n d 1 0 0 % ) e v e n if it is th e lik e lie s t o u tc o m e fo r 1 0 0 c o in to s s e s . T h e p ro b a b ility o f a n e x a c t 5 0 % o u tc o m e d e c re a s e s fu r th e r w ith th e n u m b e r o f c o in to s s e s . In th e fig u re b e lo w th e b in o m ia l d is trib u tio n is p lo tte d fo r 1 0 0 , 3 0 0 a n d 6 0 0 c o in to s s e s .A 5 0 % o u tc o m e c o rr e s p o n d s to e x a c tly 5 0 , 1 5 0 , 3 0 0 h e a d s . It is o b v io u s fro m th e fig u r e th a t th e p ro b a b ility o f th is o u tc o m e d e c re a s e s s te a d ily w ith th e n u m b e r o f c o in to s s e s . A t th e s a m e tim e th e p ro b a b ility h is to g ra m fo r p e rc e n ta g e s (n o t p lo tte d ) w ill , w ith a n in c re a s e in th e n u m b e r o f c o in to s s e s , c o n c e n tra te m o r e a n d m o re a r o u n d th e e x p e c te d v a lu e o f 5 0 % . In th e in fin ite lim it th e p r o b a b ility o f a n o u tc o m e w ith 5 0 % h e a d s w ill b e z e ro , b u t th e p ro b a b ility w ill b e o n e th a t t h e p e r c e n ta g e o f h e a d s w ill lie w ith in a n in te rv a l [5 0 % ‐ ε , 5 0 % + ε ] th a t c a n b e m a d e a s s m a ll a s d e s ir e d a s lo n g a s ε > 0 . R e v ie w E x e r c is e 1 6 .9 ( m e a n t to illu s tr a te th e la w o f la rg e n u m b e rs ) C h o o s e (ii) . W ith m o re d ra w s , th e p e rc e n ta g e o f r e d s is lik e ly to b e c lo s e r to th e p e r c e n ta g e in th e b o x ( i.e . p e rc e n ta g e > 5 0 % ) . A s th e re a re m o r e re d m a r b le s th a n b lu e m a r b le s in th e b o x m o r e d ra w s w ill in c re a s e th e p r o b a b ility o f w in n in g . T h is re v ie w e x e rc is e a n a lo g o u s to th e r e v ie w e x e rc is e 1 6 .4 b . R e v ie w E x e r c is e 1 7 .5 ( m e a n t to illu s tr a te th e b e h a v io r o f th e s ta n d a rd e r ro r in i.i.d . p ro c e s s e s ) T h e s ta n d a r d e rro r o f th e p ro b a b ility h is to g ra m in c re a s e s w ith th e s q u a re ro o t o f n . T h e re fo re , o n e c a n e x p e c t la rg e r w in s a n d la rg e r lo s s e s if th e n u m b e r o f d ie in c r e a s e s . T h e e x p e c te d v a lu e o f p r o fits (a s s u m in g o n e a lw a y s b e ts o n th e b e s t g u e s s , i.e . th e e x p e c te d v a lu e o f n x 3 .5 ) s ta y s c o n s ta n t, b u t th e v a ria b ility o f p ro fits in c re a s e s . A r is k a v e rs e p e rs o n w o u ld p r e fe r 5 0 th r o w s , a ris k lo v in g p e rs o n w o u ld p re fe r 1 0 0 th r o w s , a n d a r is k n e u tr a l p e rs o n w o u ld b e in d iffe re n t. R e v ie w E x e r c is e 1 7 .1 0 ( m e a n t to illu s tra te th e b e h a v io r o f e x p e c te d v a lu e a n d s ta n d a rd e r ro r in i.i.d . p ro c e s s e s ) S ta te m e n t ( iii) is fa ls e . T h e s ta n d a rd e rro r d o e s n o t n o t g o u p b y th e fu ll fa c to r o f 2 , b u t o n ly 2 ≈ 1 .4 . U n le s s th e re is s o m e th in g ra th e r s tra n g e a b o u t th e b o x , th e c h a n c e th a t th e s u m is b e tw e e n 7 0 0 a n d 9 0 0 w ill b e h ig h e r th a n 75%. T h e e x p e c te d v a lu e o f i.i.d . p r o c e s s e s is p r o p o r tio n a l to th e n u m b e r o f r e p e titio n s , i.e . it in c r e a s e s w ith n . T h e s ta n d a r d e r r o r o f i.i.d . p r o c e s s e s is p r o p o r tio n a l to th e s q u a r e r o o t o f th e n u m b e r o f r e p e titio n s , i.e . it in c r e a s e s w ith √ n . T h is is n o t to b e c o n fu s e d w ith th e b e h a v io r o f th e a v e r a g e a n d th e s ta n d a r d d e v ia tio n u n d e r lin e a r tr a n s fo r m a tio n s . R e v ie w E x e r c is e 1 7 .1 2 ( m e a n t to illu s tra te th e c o n tra c t b e tw e e n o b s e rv e d a n d e x p e c te d v a lu e ) T h e a v e ra g e a n d s ta n d a r d d e v ia tio n o f th e b o x , i.e . th e e x p e c ta tio n o f th e p r o b a b ility d e n s ity fu n c tio n o f th e ra n d o m v a ria b le X i th a t u n d e rlie s th e i.i.d . p r o c e s s , a re 4 a n d th e s ta n d a r d d e v ia tio n is 2 .T h e re fo r e , i n a ll th re e c a s e s , th e e x p e c te d v a lu e is 4 0 0 a n d th e s ta n d a rd e rro r is 2 0 : th e e x p e c te d v a lu e a n d s ta n d a rd e rro r a re c o m p u te d fro m th e b o x , n o t th e d ra w s . T h e o b s e rv e d v a lu e s o f th e s u m a re 4 3 1 , 3 8 6 , 4 1 7 . T h e c h a n c e e rro rs a re 3 1 , − 1 4 , a n d 1 7 , r e s p e c tiv e ly ( o b s e rv e d v a lu e o f s u m = e x p e c te d v a lu e + c h a n c e e r ro r ). R e v ie w E x e r c is e 1 8 .3 ( m e a n t to u n d e rlin e th e fa c t th a t th e n o r m a l a p p ro x im a tio n o f p ro b a b ility h is to g ra m s is ju s t a n a p p ro x im a tio n ) T h e c h a n c e th a t th e s u m w ill b e in th e in te rv a l fro m 1 0 to 2 0 in c lu s iv e e q u a ls th e a re a u n d e r th e p r o b a b ility h is to g ra m b e tw e e n 9 .5 a n d 2 0 .5 . T h e n o r m a l c u rv e is a n a p p r o x im a tio n , th e p ro b a b ility h is to g r a m c o rre s p o n d s to th e tru e d is trib u tio n . Y o u s ta rt a t 9 .5 o n th e le ft to c a tc h th e b lo c k o v e r 1 0 , re p r e s e n tin g th e c h a n c e th a t th e s u m w ill b e 1 0 . L ik e w is e , y o u s to p a t 2 0 .5 o n th e rig h t to c a tc h th e b lo c k o v e r 2 0 . T h e fig u re b e lo w c o rre s p o n d s to 2 5 d ra w s fro m th e b o x |0 |0 |0 |0 |0 |0 |0 |0 |0 |1 | a n d is m e a n t to illu s tra te th e p o in t fo r th e in te rv a l [1 ,3 ] R e v ie w E x e r c is e 1 8 .5 ( m e a n t to s h o w th e d iffe re n c e s b e tw e e n th e b o x h is to g ra m ( = p ro b a b ility d e n s ity fu n c tio n o f th e ra n d o m v a ria b le th a t u n d e rlie s th e i.i.d . p ro c e s s , th e p ro b a b ility h is to g ra m o f th e s u m o v e r m a n y b o x d r a w s , a n d th e p ro b a b ility h is to g ra m o f th e p ro d u c t o v e r m a n y b o x d ra w s ) (i) is th e p r o b a b ility h is to g ra m fo r th e s u m ; it re s e m b le s th e n o rm a l c u rv e a n d w e k n o w th a t th e s u m o v e r m a n y b o x d ra w s r e s e m b le s th e n o rm a l c u rv e a fte r a s u ffic ie n t n u m b e r o f d ra w s ; m o re o v e r , th e n u m b e r o f v a lu e s is to o h ig h a n d th e s h a p e to o d iffe re n t fro m th e p ro b a b ility d e n s ity fu n c tio n th a t c o rr e s p o n d s to th e b o x ( ii) is th e p ro b a b ility h is to g ra m fo r th e p ro d u c t ; it d o e s n o t re s e m b le th e n o r m a l c u rv e a n d th e n u m b e r o f v a lu e s is to o h ig h a n d th e s h a p e to o d iffe r e n t fro m th e p ro b a b ility d e n s ity fu n c tio n th a t c o rr e s p o n d s to th e b o x (iii) is th e h is to g ra m fo r th e n u m b e r s d ra w n ; th e h is to g ra m s u g g e s ts th a t th e c o rre s p o n d in g ra n d o m v a ria b le h a s 3 p o s s ib le v a lu e s ; th e h is to g r a m fo r th e ra n d o m v a ria b le th a t is d e fin e d th ro u g h th e s u m o v e r 2 5 re p e titio n s , a n d th e h is to g ra m fo r th e ra n d o m v a ria b le th a t re s u lts fro m th e p ro d u c t o v e r 2 5 re p e titio n s h a v e b o th m a n y m o re o u tc o m e s (th e p r o b a b ility h is to g ra m is d ra w n in c o rre c tly ; it s h o u ld lo o k lik e th is : R e v ie w E x e r c is e 1 8 .9 ( m e a n t to illu s tr a te p o s s ib le s h o r tc o m in g s o f th e n o r m a l a p p ro x im a tio n fo r s u m s th a t re s u lt fr o m i.i.d . p ro c e s s e s ) T h e a v e ra g e o f th e b o x is 1 / 1 0 0 a n d th e s ta n d a rd d e v ia tio n is 0 .9 9 5 . If th e b o x is u s e d in a n i.i.d . p ro c e s s th e e x p e c ta tio n o f th e s u m o v e r th e re p e titio n s is 1 a n d th e s ta n d a rd e rro r is 9 .9 5 . (a ) tr u e ; s e e a b o v e (b ) fa ls e ; w o u ld b e tr u e if th e n o rm a l d is trib u tio n h o ld s , b u t th is is n o t s o fo r th is s p e c ific b o x m o d e l; th e n u m b e r o f d ra w s is n o t e n o u g h to u s e th e n o r m a l a p p ro x im a tio n ; s e e e x e rc is e 1 8 . 6 , p .3 2 4 . E x c e l M o n te C a r lo E x e r c is e ( m e a n t to illu s tr a te th e d iffe re n c e b e tw e e n in d iv id u a l re a liz a tio n s a n d p ro b a b ility h is to g r a m s o f ra n d o m v a r ia b le s ; fu r th e r m e a n t to e m p h a s iz e th e d iffe re n c e b e tw e e n m a n y r e p e titio n s o f a s p e c ific p r o c e s s – i.i.d . o r n o t ‐ a n d a la rg e r n u m b e r o f r e p e titio n s in i.i.d . p r o c e s s e s ; fin a lly m e a n t to s h o w th e c o n te n t o f th e la w o f la rg e n u m b e rs a n d th e c e n tra l lim it th e o re m ) T h e g ra p h s in th e fig u re b e lo w c a n b e u s e d to illu s tra te th e n o r m a l a p p ro x im a tio n o f p ro b a b ility h is to g ra m s (c e n tra l lim it th e o re m ) th e la w o f a v e ra g e s ( la w o f la r g e n u m b e rs ), a n d th e fre q u e n tis t a p p ro a c h to p r o b a b ility . T h e ra n d o m v a ria b le X n = h ( X 1 , .. , X n ) is c h a ra c te r iz e d b y a p r o b a b ility d e n s ity fu n c tio n . O n e c a n s im u la te re p e titio n s o f th e i.i.d . p r o c e s s th ro u g h M o n te C a r lo s im u la tio n s . T h e re la tiv e fr e q u e n c ie s o f th e s e re p e titio n s a p p ro x im a te th e p ro b a b ility h is to g ra m o f X n a n d , in th e in fin ite lim it, w i ll b e id e n tic a l w ith it. In th e fre q u e n tis t in te rp re ta tio n o f p r o b a b ility p ro b a b ilitie s e q u a l re la tiv e fr e q u e n c ie s a fte r a n in fin ite n u m b e r o f re p e titio n s a n d c o m e c lo s e to th e m a fte r a fin ite n u m b e r o f re p e titio n s . Density histograms for i.i.d. process with probability density function fX(x); histograms summarize 1000 Monte Carlo realizations of random variable Xn=X1+..+Xn (left side) and X’n=(X1+..+Xn)/n (right side); 4 n values are depicted, namely n = 50, 100, 250, 500 (from top to bottom) n Average SD Max Min 50 576.15 110.70 920 228 100 1147.16 159.75 1626 682 250 2881.05 254.48 3882 1976 1000 5752.80 360.03 6702 4706 50 23.26 4.43 37.84 8.56 100 11.52 1.66 16.24 5.48 250 11.54 1.02 14.66 8.23 1000 11.50 0.73 13.66 9.48 T h e re fo re , th e h is to g ra m s in th e fig u re a p p r o x im a te th e p ro b a b ility h is to g ra m s o f th e ra n d o m v a ria b le s X n = h (X 1 ,..,X n ). T h e 4 g ra p h s o n th e le ft ‐ h a n d s id e a p p ro x im a te th e p ro b a b ility h is to g r a m o f th e r a n d o m v a ria b le X n = X 1 + ..+ X n a n d th e 4 g ra p h s o n th e rig h t ‐ h a n d s id e a p p ro x im a te th e p r o b a b ility h is to g r a m o f th e ra n d o m v a ria b le X ’ n = (X 1 + ..+ X n )/ n . A s th e ra n d o m v a ria b le s r e s u lt fro m i.i.d . p ro c e s s e s o n e c a n c a lc u la te th e ir e x p e c te d v a lu e s a n d s ta n d a r d e r r o r s e a s ily . E ( X n ) = nX = n ⋅ 11.5 SE ( X n ) = n ⋅ 15.9 E ( X 'n ) = X = 11.5 15.9 SE ( X 'n ) = n € If o n e lo o k s a t th e ta b le b e lo w th e fig u re o n e c a n s e e th a t th e a v e ra g e s a n d s ta n d a rd d e v ia tio n s o f th e 8 M o n te C a rlo g ra p h s a r e v e ry c lo s e to th e e x p e c ta tio n s a n d s ta n d a rd e rr o rs o f th e p r o b a b ility d e n s ity fu n c tio n s . T h e n o r m a l a p p r o x im a tio n o f h is to g r a m s m a k e s it p o s s ib le to s a y th a t, fo r a s u ffic ie n tly la r g e n u m b e r n , th e p ro b a b ility h is to g ra m o f X n c a n b e a p p ro x im a te d b y a n o r m a l d is tr ib u tio n w ith E (X n ) = n x 1 1 .5 a n d S E (X n ) = √ n x 1 5 .9 . T h e d e n s ity h is to g r a m s in th e 4 le ft ‐ h a n d e d g ra p h s re s e m b le in d e e d th e n o rm a l c u r v e e v e n th o u g h m a jo r s p ik e s in th e c u rv e a r e a p p a r e n t. T h e la w o f la r g e n u m b e r s p re d ic ts th a t th e p ro b a b ility h is to g r a m o f X ’ n w ill, w ith in c re a s in g n , c o n c e n tra t e m o re a n d m o re o n a n a rro w in te rv a l a r o u n d 1 1 .5 . T h is is in d e e d w h a t s e e m s to h a p p e n in th e 4 rig h t s id e d ‐ g ra p h s . R e v ie w E x e r c is e 2 1 .1 ( m e a n t to illu s tr a te th e b a s ic s o f in fe re n c e fro m th e s a m p le to th e p o p u la tio n ) T h e n u m b e r in th e s a m p le w h o a p p ro v e o f th e M a y o r is lik e th e s u m o f 1 0 0 0 d ra w s m a d e a t ra n d o m w it h o u t re p la c e m e n t fr o m a b o x w ith a tic k e t fo r e a c h re g is te re d v o te r in th e to w n ; th e tic k e t is m a r k e d “1 ” if th e v o te r a p p ro v e s o f th e M a y o r a n d “0 ” o th e rw is e . T h is is ju s t a n o th e r e x a m p le h o w c h a n c e v a r ia b ility in a re a l w o rld p ro c e s s c a n b e d e s c r ib e d b y a b o x m o d e l/ p ro b a b ility m o d e l. R e v ie w E x e r c is e 2 1 .2 ( m e a n t to illu s tr a te th e b a s ic s o f in fe re n c e fro m th e s a m p le to th e p o p u la tio n ) T h e s a m p le is lik e 5 0 0 d ra w s fro m a b o x w ith 2 5 ,0 0 0 tic k e ts ; e a c h tic k e t is m a r k e d 1 ( h a s a c o m p u te r) o r 0 (d o e s n o t h a v e a c o m p u te r ). T h e n u m b e r o f s a m p le h o u s e h o ld s w ith c o m p u te rs is lik e th e s u m o f th e d r a w s . T h e fra c tio n o f 1 ’s in th e b o x is u n k n o w n , b u t c a n b e e s tim a te d b y th e fr a c tio n in th e s a m p le , a s 2 3 9 / 5 0 0 = 0 .4 7 8 . O n th is b a s is , th e S D o f th e b o x is e s tim a te d a s √ 0 .4 7 8 × 0 .5 2 2 ≈ 0 .5 0 . T h e S E fo r th e n u m b e r o f s a m p le h o u s e h o ld s w ith c o m p u te r s is e s tim a te d a s √ 5 0 0 × 0 .5 0 ≈ 1 1 , a n d 1 1 o u t o f 5 0 0 is 2 .2 % . T h e a n s w e rs to (a ) a n d (b ) fo llo w im m e d ia te ly . (a ) T h e p e rc e n ta g e o f h o u s e h o ld s in th e to w n w ith in te rn e t a c c e s s is e s tim a te d a s 4 7 .8 % , a n d th e e s tim a te is lik e ly to b e o ff b y 2 .2 % o r s o . (b ) T h e 9 5 % ‐ c o n fid e n c e in te r v a l is 4 7 .8 % ± 4 .4 % . In th e fre q u e n tis t in te rp r e ta ti o n o f p r o b a b ility th is m e a n s th a t th e c o n fid e n c e in te r v a l c o v e r s th e tru e p o p u la tio n p a ra m e te r in 9 5 % o f th e s a m p le s th a t a r e d ra w n ra n d o m ly fr o m th e p o p u la tio n . ...
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