# UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS

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1 UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Postpoed exam: ECON430 Statstcs Date of exam: Jauary 0, 0 Tme for exam: 09:00 a.m. :00 oo The problem set covers 5 pages Resources allowed: All wrtte ad prted resources, as well as calculator, s allowed The grades gve: A-F, wth A as the best ad E as the weakest passg grade. F s fal. Problem Let the radom varables (rv s), X ad Y, be jotly uformly dstrbuted sde the tragle ABC show fgure. Ths meas that the jot probablty desty fucto (pdf) s gve by c.e., costat for ( x, y) sde the tragle ABC fgure f ( x, y) 0 wheever ( xy, ) s outsde the tragle where c s a costat determed such that f ( x, y ) s a pdf. Note that X vares the terval (, ), whle Y vares (0, ). Fgure The area where ( XY, ) ca have observatos.

2 A. Expla why the costat f must be equal to c. [Ht: Remember that the area of a tragle s equal to the basele tmes the heght dvded by. Also, remember that the volume of a box s the area of the bottom (base) tmes the heght of the box. The box ca have ay shape as log as the sze ad shape of the top s equal to the sze ad shape of the bottom of the box. Note that, because of ths, the probablty that ( XY, ) falls sde ay rego R that les sde the tragle fgure, must smply be equal to the area of R sce the box betwee R ad the pdf has the same heght everywhere. The volume of the box (bottom area tmes heght) must the be equal to the sze of the bottom area. For example, the probablty that ( XY, ) falls the postve part of the tragle fgure (.e., wth R equal to the tragle delmted by Orgo ad B,C), must be ½ sce the box above R has heght ad the area of R s. Hece the probablty, whch s the volume betwee R ad the pdf, s also. ] B. () For fxed y, show, for example by drawg a fgure, that f ( x, y) 0 whe y x y. () Show that the margal pdf for X s gve by x for x 0 f X ( x) x for 0 x 0 otherwse () Show that the margal pdf for Y s gve by f ( y) ( y) for 0 y Y C. Fd () PY ( 0.5), () PY ( 0.5), ad () P( Y X ). [Ht for (): Draw a fgure lke fgure ad dcate the part of the tragle ABC where y x. ] D. () Show that the codtoal dstrbuto of X, gve that Y y s fxed, s uform over the terval from ( y) to y. () Show that, for ay teger r,,3,

3 r ( y) ( r ) r r y ( ) E X () Fd expressos for the regresso fucto, E( X y ), ad the codtoal varace, var( X y ). (v) Are X ad Y depedet rv s? State a reaso for your aswer. E. Calculate the correlato coeffcet betwee X ad Y. E( XY ) E Y E( X Y) etc ] [Ht: Use the law of total expectato to fd Problem Let the cotuous radom varable (rv), X, have the cumulatve dstrbuto fucto (cdf) gve by for x 0 Fx ( ) ( x) 0 for x 0 where, are parameters such that 0 ad 0. Ths s a specal verso of the pareto dstrbuto, whch we may call the pareto(, ) dstrbuto. A terestg property of ths dstrbuto s that the expected value, EX ( ), does ot exst (.e., EX ( ) ) f 0. O the other had, f, t ca be show (whch you do ot eed to do here) that EX ( ) ( ). A. () Show that the probablty desty fucto (pdf) of X s gve by for x 0 f( x) ( x) 0 otherwse () The cdf Fx ( ) s ot dfferetable for x 0. Does t matter for the dstrbuto of X f we defe the pdf x 0 as, e.g., f (0) 0 or as f (0)? Gve a reaso for your aswer. () Suppose X s pareto(, ) dstrbuted. Show that X s pareto(,) dstrbuted. 3

4 B. A surace compay assumes that the clam szes for a partcular o-lfe surace category are dstrbuted accordg to the pareto(, ) dstrbuto, ad wats to estmate, based o a d sample cosstg of the 00 observatos show table. Table 00 surace clams. (Ut of measuremet, NOK 00.) Thus, the observatos table are cosdered as observatos of rv s, X, X,, X, whch are assumed to be d wth each X beg pareto(, ) dstrbuted. () Derve the the equatos ecessary to determe the maxmum lkelhood estmators for ad. Do ot try to solve the equatos (they requre teratos). () The mle estmates, whch requre teratos, tur out to be ˆ obs 0.50 ad ˆ obs 0.43, where the dex obs dcates the observed value. Show how the value ˆ obs 0.50 follows gve the formato table. Table Statstcs Observed value ˆ l( x ˆ ) ˆ x () Usg asymptotc theory for mle s, combed wth Slutsky s lemma, we obta a approxmate dstrbuto for the vector ( ˆ, ˆ)', for fxed ad large ( 00 should be suffcet), as a bvarate ormal dstrbuto wth kow covarace matrx, I.e., depedet ad detcally dstrbuted. 4

5 ˆ approxmately ~ N, ˆ Usg ths result set up a approxmate % level test-crterum for the ull-hypothess, H : agast H : 0. Perform the test ad commet o the result. C. Show that the Fsher formato matrx for oe observato s gve by ( ) I(, ) ( ) ( ) [Ht: You may eed the followg results (that you do t eed to prove here): X E X ad X E X ( )( ) that are vald for all 0. ] D. () Develop the asymptotc covarace matrx (gve a large fxed ) for the mle vector ( ˆ, ˆ)', expressed by,, ad. [Ht: Remember that the verse of a o-sgular symmetrc x- matrx s gve by a c b c, c b D c a where D ab c s the determat of a c c b ] () Derve a formula for the asymptotc correlato coeffcet betwee the mle s ˆ ad ˆ ad estmate t from the data usg cosstet estmators. 5

### UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS

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