12 Basic Integration in R

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1 14.102, Mt for Economists Fll 2004 Lecture Notes, 10/14/2004 Tese notes re primrily bsed on tose written by Andrei Bremzen for in 2002/3, nd by Mrek Pyci for te MIT Mt Cmp in 2003/4. I ve mde only minor cnges to te order of presenttion, nd dded few sort emples. Te usul disclimer pplies; questions nd comments re welcome. Ntn Brczi nb@mit.edu 12 Bsic Integrtion in R A function on bounded intervl [, b] is piece-wise continuous if it is continuous everywere ecept on finite number of points in I ndttteverypointwere it is not continuous it dmits finite left nd rigt limits. Definition 197 For piecewise continuous functions on intervl I, we define for ny nd b in I ( <b): f() = b lim n + n By definition: R b f() = R b f() nx k=1 f( + k (b )) n Te geometric interprettion of tis integrl (te Riemnn integrl) is te re under te curve. Look t te nottion. Cn sow tt suc limit eists. Tere re problems wit tis notion of te integrl; too often it is not well defined. It is possible to etend te definition of Riemnn integrl to broder clss of functions. One suc useful generliztion is Lebesgue integrl bsed on mesure teory. First note tt mesure teory generlizes te notions of lengt, re, nd volume; you will need dose of it wile studying probbility teory for sttistics nd econometrics. Dividing [, b] into ny mesurble sets nd using mesures of set insted of lengts of intervls we cn reproduce te bove definition nd bsiclly construct te Lebesgue integrl. Te pyoff from tis more elborte construction is tt te limit often eist, nd ence ny well-beved function is mesurble nd dmits n integrl (you my encounter some mesurebility problems in te stocstic dynmic progrmming toug). Note, n importnt property of te Lebesgue integrl is tt te integrl over sets of mesure 0 s vlue 0 wic is not necessrily te cse for te Riemnn construction! Oterwise Lebegue s s pretty muc te sme properties s 72

2 Riemnn s (nd it gives te sme number s Riemnn s wenever Riemnn s eists). A key emple probbility density f() nd cumultive distribution F (). Proposition 198 Useful properties of te integrl: if f 0 nd b, ten R b f 0 (f + g) =R b f + R b g R λf() = λ R f() f() = R c f() + R b c f() if f() g() for ll [, b] ten R b f() R b g() f() R b f() f() sup f() b f()g() sup f() R b g() Useful primitives: (1/) =ln(b) ln() ep() =ep(b) ep() tα dt = bα+1 α+1 α Fundmentl Teorem of Clculus () Let f be integrble on n intervl I, ndlet I. LetF () = R. If f is continuous t ten F is differentible t nd F 0 () d Z = f(). (b) Suppose tt F is differentible on n intervl I nd tt F 0 = f is integrble. Ten Proof. () f() = F (b) F () for, b I. F 0 () = F ( + ) F () = R + R = 73 R +

3 Since, f is continuous t so for ny ε>0 if is smll enoug ten we ve f ( + ) f () <ε nd tus, if we define s = : f( ) f() =sup 0 [,+] f( 0 ) f(), we ve F ( + ) F () f () R + = f () f( ) f () < ε = ε s 0 we cn tke ε 0 nd tus F 0 F ( + ) F () () = lim = f (). 0 R (b) Let s consider only te esy cse wen f is continuous. Denote G () = nd notice tt by () G0 = f. Hence, F nd G ve sme derivtive, nd (F G) 0 =0nd tus F G is constnt, denote it c. Tus, F (b) F () =(G (b)+c) (G ()+c) = G (b) G () =. 13 Cnge of vribles nd Integrtion by prts Teorem 199 Cnge of vribles: Let J 1 nd J 2 be intervls (wit more tn one point):let f : J 1 J 2 nd g : J 2 R continuous. Assume tt f is differentible nd f 0 continuous. Ten for ny, b in J 1 g(f())f 0 () = Z f(b) f() g(u)du (18) Proof. Let G be primitive of g. TenG f is te primitive of g(f())f 0 () by te cin rule. Bot terms in (18) re equl to G(f(b)) G(f()). Emple: I = R 1 0 2e2, sy u = e 2 nd du =2e 2, I = R e du = e 1. 1 Be fmilir wit te following two importnt specil cses: f(t + α)dt = f(αt)dt = +α +α Z αb α f(u)du f(u) α du Proposition 200 Integrtion by prts: Suppose F nd G re differentible on [, b]. SupposeF 0 = f nd G 0 = g. re continuous. Ten: f(t)g(t)dt =[F (b)g(b) F ()G()] F (t)g(t)dt You cn obtin tis formul quickly by noticing tt [FG] 0 = fg+ Fg. Eercise 201 Find R log(t)dt. 74

4 13.1 Differentition Under te Integrl Sign Often we encounter situtions under wic we wis to intercnge te order of integrtion nd differentition. Proposition 202 Leibniz rule: wit respect to, ten: If f(t, ), (), nd b() re differentible d () () f(t, )dt = () () f(t, ) dt + b 0 ()f(b(),) 0 ()f((),) Note tt if () nd b() re constnt, we ve specil cse: d f(t, )dt = f(t, ) dt Notice tt tis question relly comes down to wen it is justifible to ecnge te order of integrtion nd limit, since te derivtive is prticulr kind of limit. A full tretment of tis question requires bit of mesure teory, wic we won t go into ere. However, couple importnt results cn be presented, ll of wic re vritions on Lebesgue s Dominted Convergence Teorem (see Rudin; see lso section 2.4 of Csell nd Berger). Teorem 203 Suppose te function (, y) is continuous t y 0 for ec, nd tere eists function g () stisfying 1. (, y) g() for ll nd y, 2. R g() < Ten lim (, y) = lim (, y) y y 0 y y 0 Te key condition is te eistence of dominting function g(), wit finite integrl, wic ensures tt te integrl of (, y) cnnot be too bdly beved. If we pply tis to te cse we re interested in, te derivtive, we ve Teorem 204 Suppose f(t, ) is differentible t = 0,ttis, f(t, 0 + ) f(t, 0 ) lim = f(t, ) 0 =0 eists for every t, nd tere eists function g(t, 0 ), for ll t nd constnt 0 > 0 suc tt 1. f(t, 0+) f(t, 0 ) g(t, 0 ), for ll t nd 0, 75

5 2. R g(t, 0) <. Ten Z d f(t, ) = =0 f(t, ) = 0 Te conditions essentilly bound vribility in te derivtive of te function; tey re similr to smootness condition clled te Lipscitz condition. Most of te pplictions of tese results wic you ll see will come in sttistics nd econometrics, were mny results in symptotic teory emining te convergence bevior of function s our dt become infinite begin wit condition bounding te vrince of te function in question. Note tt te teorem is stted for prticulr vlue of ; oftenweve functions wic re differentible over some intervl, nd te teorem olds for witin tis intervl insted of single vlue of Improper Integrls R A Remrk: improper integrl: if lim A + eists,wenoteitr + Emple : R + e rt dt =[ e rt r ] + = e r r Eercise 205 compute R + te rt dt (use n integrtion by prt) Eercise 206 compute R + 0 e t dt (use te cnge of vrible u = t) 76