D I S C U S S I O N P A P E R


 Margery Warner
 4 years ago
 Views:
Transcription
1 I N S T I T U T D E S T A T I S T I Q U E B I O S T A T I S T I Q U E E T S C I E N C E S A C T U A R I E L L E S ( I S B A UNIVERSITÉ CATHOLIQUE DE LOUVAIN D I S C U S S I O N P A P E R 2012/27 Worstcase actuarial calculatios cosistet with sigle ad multipledecremet life tables CHRISTIANSEN, M.C. ad M. DENUIT
2 WORSTCASE ACTUARIAL CALCULATIONS CONSISTENT WITH SINGLE AND MULTIPLEDECREMENT LIFE TABLES MARCUS C. CHRISTIANSEN Istitute of Isurace Sciece Uiversity of Ulm D Ulm, Germay MICHEL M. DENUIT Istitut de Statistique, Biostatistique et Scieces Actuarielles Uiversité Catholique de Louvai B1348 LouvailaNeuve, Belgium Jue 15, 2012
3 Abstract The preset work complemets the recet paper by Barz ad Müller (2012. Specifically, upper ad lower bouds are derived for the force of mortality whe oeyear death probabilities are give, assumig a mootoic or covex shape. Based o these bouds, worstcase scearios are derived depedig o the mix of beefits i case of survival ad of death comprised i a specific isurace policy. Key Words: life isurace, multistate models, firstorder life tables.
4 1 Itroductio I actuarial studies, life tables are based upo a aalytical framework i which death is viewed as a evet whose occurrece is probabilistic i ature. I the sigledecremet case, the differet causes of death are ot distiguished. Life tables create a hypothetical cohort of, say, 100,000 persos at age 0 ad subject it to allcause agespecific oeyear death probabilities q x (the umber of deaths per 1,000 or 10,000 or 100,000 persos of a give age observed i a give populatio. I doig this, researchers ca trace how the 100,000 hypothetical persos (called a sythetic cohort would shrik i umbers due to deaths as the group ages. Deotig as l x the expected umber of survivors at age x, we have the recurrece relatio l x+1 = l x (1 q x, startig from a fixed umber l 0 of ewbors. Here, p x = 1 q x is the oeyear survival probability at age x. Multiple decremet life tables distiguish betwee differet causes of death (traffic accidet, suicide, cacer, etc.. They are useful whe the beefit i case of death may deped o the cause of death (typically, higher beefit i case of accidetal death, or o beefit i case death is due to some particular cause. Such life tables display the expected umber l x of idividuals alive at age x together with the expected umber of death due to cause 1 to m, say. The cause of death is defied as the disease or ijury that iitiated the sequece of evets leadig directly to death. It is selected from the Iteratioal Classificatio of Diseases (ICD, for istace. Allcause death rates are the replaced with causespecific death rates. The 4 leadig causes of death are geerally heart disease, stroke, accidets, ad cacer. See, e.g., Jemal et al. (2005 for more details. As explaied above, a life table ca be see as a sequece of oeyear survival probability p x or death probability q x idexed by iteger age x ragig from birth (or some other iitial age α to a ultimate age ω. The kowledge of the life table itself does ot characterize the probability distributio of a radom lifetime T coformig to it. For iteger x, we have Pr[T > x] = lx l 0 but these probabilities are ot kow for fractioal ages. This is why actuaries ad demographers developed socalled fractioal age assumptios allowig to iterpolate survival fuctios betwee cosecutive values lx l 0 ad l x+1 l 0 for iteger x = α, α + 1,..., ω 1. Some recet developmets i this regard are Joes ad Mereu (2000, 2002, Frostig (2002, 2003, ad Hossai (2011. Barz ad Müller (2012 ordered the differet fractioal age assumptios ad built radom variables that are stochastically larger tha other lifetimes subject to icreasig force of mortality ad give oeyear survival probabilities. Assumig that the preset value of beefits is a mootoic fuctio of policyholder s remaiig lifetime (as i the whole life isurace cover or i a auity cotract, for istace, Barz ad Müller (2012 derived bouds o actuarial quatities uder the assumptio that the force of mortality is icreasig ad coheret with a give sequece of oeyear survival probabilities. The icreasigess of the forces of mortality seems to be i lie with observatios (at least at adult ages. Typically, oeyear death probabilities are relatively high i the first year after birth, declie rapidly to a low poit aroud age 10, ad thereafter rise, i a roughly expoetial fashio, before deceleratig (or slowig their rate of icrease at the ed of the life spa. Except aroud the accidet hump due to violet deaths (suicide ad traffic accidets, maily at youg adult ages ad some cotroversy about mortality of supercetearias, it is reasoable to assume that allcause forces of mortality icrease mootoically with age. 1
5 The preset paper aims to relax the two mai assumptios made i Barz ad Müller (2012: icreasig death rates ad isurer s liabilities mootoic with respect to the policyholder s remaiig lifetime. Ideed, the majority of isurace products mix beefits i case of death ad i case of survival so that the preset value of beefits may ot be mootoic i the policyholder s remaiig lifetime. Moreover, whe multiple causes of death are cosidered, the mootoicity coditio imposed to death rates does ot apply to certai causes (like suicide or traffic accidets, for istace while it remais plausible for others (like cacer, for example. The preset paper relaxes these rather strog assumptios ad derives worstcase scearios i lie with a specific life table ad give the mix comprised i a specific isurace product. Precisely, we show how the worstcase techiques of Christiase ad Deuit (2010 ad Christiase (2010 ca be used to obtai bouds where the methods of Barz ad Müller (2012 are ot applicable. The ext table summarizes the extesios to Barz ad Müller (2012 derived i the preset paper, stressig the techical differeces arisig betwee the two studies: Barz ad Müller, 2012 preset paper patter of states sigledecremet sigle or multiplealivedead model decremet model cotract desig liability mootoe with all mix of survival respect to lifetime ad death beefits bouds (partially sharp ot sharp but accurate eough for practical purposes The paper is orgaized as follows. Sectio 2 presets the multistate model used i this paper ad makes the lik with the approach of Barz ad Müller (2012. Sectio 3 cosiders the 2state alivedead model, or sigledecremet case whereas Sectio 4 exteds the results to the multipledecremet case. Based o the mootoicity or covexity of forces of mortality, bouds are derived o every age iterval. Cosiderig the sig of the sumatrisk, worstcase sceario ad bestcase sceario for the reserves correspod to these lower or upper bouds. Some umerical illustratios demostrate the accuracy of these values. Note that lapses ca also be accouted for by addig a state m + 1 correspodig to the exit from the portfolio, with appropriate lapse rates. This is aother major improvemet to Barz ad Müller ( Survival model 2.1 Multipledecremet model Assume that the history of each policyholder is described by a rightcotiuous Markovia process {X t, t 0} with state space {0, 1,..., m}, where 0 is the iitial state alive ad 1 to m correspod to m differet causes of death. As metioed i the itroductio, oe of the states 1 to m may correspod to lapse or surreder. Here, time t measures the seiority of the policy (i.e., the time elapsed sice policy issue. Policyholder s age at policy issue is deoted as x, so that the age at time t is x + t. Heceforth, all quatities are idexed by t (the correspodig age beig x + t. 2
6 The probability distributio of the Markov process {X t, t 0} is uiquely described by the forces of mortality µ 0j, j = 1,..., m, defied as µ 0j (t = lim t 0 Pr[X t+ t = j X t = 0]. t We cosider a geeral life isurace cotract icludig beefits i case of survival (sojour i state 0 ad i case of death (trasitio from state 0 to some state 1 to m. We write B 0 (t for the aggregated survival beefits mius premiums o [0, t] ad b 0j (t for the beefit paymets that are due i case of a death due to cause j occurrig at time t (b 0j may also represet the surreder value. For mathematical techical reasos we assume that the fuctios t B 0 (t ad t b 0j (t, j = 1,..., m, have fiite variatio o compacts ad that B 0 (t is rightcotiuous. The cotract termiates at time ω x = ω x <, at the latest. The policyholder s remaiig lifetime T ca be defied by meas of the process {X t, t 0} by T = sup{t 0 X t = 0}. I words, T is the istat whe the process {X t, t 0} jumps from state 0 to oe of the absorbig states 1 to m. Let ϕ(t be the iterest itesity (or spot rate. The preset value of future beefits mius premiums for a policyholder alive at time t is the give by A(t = (t,t ( exp s 0 ϕ(udu db 0 (s + The socalled prospective reserve at time t i state 0 is defied as 2.2 Sigledecremet model m j=1 V (t = E[A(t X t = 0] = E[A(t T > t]. ( T exp ϕ(udu b 0j (T. 0 I the twostate model with state space {0, 1}, 0= alive ad 1= dead, Barz ad Müller (2012 assumed that the preset value of future beefits mius premiums at time 0 is of the form f(t, where f is a mootoe fuctio. By choosig B 0 (t = 0 ad ( t b 01 (t = exp ϕ(sds f(t 0 we obtai A(0 = f(t, which meas that we ca itegrate the modelig framework of Barz ad Müller (2012 ito our framework. Note that f has fiite variatio o compacts sice it is mootoe. 3
7 3 Worstcase sceario i the sigledecremet model 3.1 Icreasig force of mortality I this sectio, we deal with a twostate space {0, 1} correspodig to the sigledecremet model. We assume that ( +1 the values of p = exp µ 01 (t dt are kow for every iteger, (3.1 the trasitio itesity µ 01 (t is mootoic icreasig. (3.2 Note that the quatities are idexed by cotract seiority, ot by age (this meas that p is the oeyear survival probability at age x +. Let M 01 be the set of all trasitio itesities µ 01 that satisfy (3.1 ad (3.2. The iformatio cotaied i (3.1 ad (3.2 does ot suffice to uiquely defie the trasitio itesity µ 01, ad so M 01 has ifiitely may elemets. We wat to fid upper ad lower bouds for the prospective reserve V (t, give that µ 01 M 01. From the two coditios (3.1(3.2 above we ca coclude that we must have for iteger times, which implies that µ 01 ( l p µ 01 ( + 1 l p 1 µ 01 ( l p for all positive itegers. Sice µ 01 (t is mootoic icreasig, we get l p 1 µ 01 (t l p +1, t + 1. (3.3 Remark 3.1. The same approach applies to decreasig forces of mortality, eve if this situatio should ot happe except i some very particular cases (life settlemets for idividuals who just udergo serious surgery, whose chaces of survival icrease as time passes, for istace. If µ 01 is decreasig, we get µ 01 ( l p µ 01 ( + 1. Thus, we ca show that, provided the force of mortality is mootoic (either decreasig or icreasig, the iequalities hold true for t + 1. mi{ l p 1, l p +1 } µ 01 (t max{ l p 1, l p +1 } 3.2 Covex or cocave forces of mortality Eve if icreasigess is geerally a reasoable assumptio for allcause death rates, let us ow derive upper ad lower bouds for other shapes of mortality itesity. This assumptio may apply to some causespecific death rates, like suicide or traffic accidets, for istace. The ext results allow us to derive bouds o covex forces of mortality. Propositio 3.2. Assume that µ 01 is covex. If µ 01 (t l p + c for t [, + 1], the µ 01 (t l p c for all t [, + 1]. 4
8 Proof. I order to get if t [,+1] µ 01 (t uder the coditio +1 µ 01 (u du = l p, the optimal shape for the covex fuctio µ 01 is a triagle with the apex poitig dowwards. As all triagles have a upper boud of l p + c ad as the areas where µ 01 is above ad below l p must be equal, the apex of the triagles caot be below l p c. Propositio 3.3. If µ 01 is covex the the iequality µ 01 (t max{ l p 1, l p +1 } holds for all t + 1. Proof. If the iterval where the covex fuctio µ 01 (t is miimal is to the left of, the µ 01 (t is icreasig for all t ad µ 01 (t l p +1 for all t [, + 1]. Aalogously, if the iterval where µ 01 (t is miimal is to the right of + 1, the µ 01 (t l p 1 for all t [, + 1]. If ξ [, + 1] is a miimum of µ 01, the µ 01 is decreasig till ξ ad icreasig from ξ o, which implies that o [, ξ] ad [ξ, + 1] the fuctio µ 01 is bouded by l p 1 ad l p +1, respectively. Combiig the two propositios above, we get the followig iequalities o a covex force of mortality: 2( l p max{ l p 1, l p +1 } µ 01 (t max{ l p 1, l p +1 } (3.4 for all t [, + 1]. Assumig that µ 01 is cocave, the same reasoig yields mi{ l p 1, l p +1 } µ 01 (t 2( l p + mi{ l p 1, l p +1 } (3.5 for all t [, + 1]. 3.3 Worstcase ad bestcase scearios for the reserve Let us start with the case where the allcause force of mortality µ 01 is icreasig. Let us deote as N 01 the set of all icreasig µ 01 that satisfy (3.3. Let l 01 (t ad u 01 (t be the lower ad upper boud (accordig to (3.3 o µ 01 (t at time t. Clearly, M 01 N 01, ad, hece, upper ad lower bouds with respect to N 01 are also upper ad lower bouds with respect to M 01. With the help of the worstcase techique of Christiase ad Deuit (2010 for mortality itesities, we obtai the followig result. Propositio 3.4. The bestcase reserve V (t = if µ01 N 01 V (t; µ 01 ad the worstcase reserve V (t = sup µ01 N 01 V (t; µ 01 uiquely solve the itegral equatios V (t =B 0 (ω x B 0 (t V (s ϕ(s ds (t,ω x] + 1 ( (b 01 (s V (s (u 01 (s + l 01 (s b 01 (s V (s (u 01 (s l 01 (s dt, 2 (t,ω x] V (t =B 0 (ω x B 0 (t V (s ϕ(s ds (t,ω x] (t,ω x] ( (b 01 (s V (s (u 01 (s + l 01 (s + b 01 (s V (s (u 01 (s l 01 (s dt. 5
9 Furthermore, applyig the results i Christiase (2010 allows us to associate V (t ad V (t to specific mortality itesities, as show ext. Propositio 3.5. Defie µ 01 (t = 1 b01 (t V (tl 01 (t + 1 b01 (t<v (tu 01 (t, µ 01 (t = 1 b01 (t V (t u 01(t + 1 b01 (t<v (t l 01(t. The, V (t = V (t; µ 01 ad V (t = V (t; µ 01 for all t. Depedig o the sig of the sumatrisk at time t (differece of the reserve ad b 01 (t, or et cost of a trasitio at time t for the isurace compay i case of a death occurrig at time t, the worstcase ad bestcase reserves the correspod to the lower or to the upper boud o the allcause force of mortality µ 01. While the bestcase ad worstcase itesities of Barz ad Müller (2012 are elemets of M 01, the itesities µ 01 (t ad µ 01 (t are either i M 01 or i N 01 \ M 01. Thus, our bouds for the prospective reserve are less tight tha the bouds give by Barz ad Müller (2012. However, Barz ad Müller (2012 assume that A(0 is mootoe with respect to the remaiig lifetime T, whereas our modelig framework does ot eed that mootoicity, allowig to study also mixed cotracts with ay combiatio of sojour beefits, premium paymets ad trasitio beefits. The same approach applies to decreasig, covex or cocave µ 01. It suffices to replace the bouds i (3.3 with those of Remark 3.1, (3.4 or (3.5. The followig umerical example compares the bouds derived from Propositios to those i Barz ad Müller (2012. Example 3.6. As i Barz ad Müller (2012, we assume that µ 01 (t = t, but that oly the p at iteger times (see (3.1 are actually kow. Cosider a wholelife isurace cover with a uit death beefit ad 6% aual iterest rate. Table 1 shows upper ad lower bouds for the prospective reserves A 30 ad A 50 for mootoic or covex force of mortality. Compared to the values reported i Barz ad Müller (2012, our bouds are less accurate but remai evertheless sharp eough for practical purposes. I the last two colums the table gives bouds for the prospective reserves of edowmet isuraces that pay a survival beefit of 2 at age 65 or a death beefit of 1, whichever occurs first. For the edowmet isuraces the preset values of future paymets are ot mootoe with respect to the policyholder s remaiig lifetime, ad so the method of Barz ad Müller (2012 is ot applicable here while the method developed i the preset paper still is. The coclusios stated before still apply i this case: the bouds are accurate eough to be iformative. 6
10 mootoicity assumptio A 30 A A E A E 50 upper boud true value lower boud covexity assumptio A 30 A A E A E 50 upper boud true value lower boud Table 1: Worstcase ad bestcase scearios for wholelife isurace cover ad edowmet isurace described i Example Worstcase sceario for multipledecremet models Let us ow exted the ideas of the previous sectio to multipledecremet models. Defie the oeyear death probability due to cause j ( +1 ξ m q (j = Pr[X +1 = j X = 0] = exp µ 0k (ηdη µ 0j (ξdξ ad the oeyear survival probability m p = 1 j=1 q (j = exp ( +1 Clearly, q = m j=1 q(j. Istead of (3.1 ad (3.2, let us ow assume that k=1 m µ 0j (ηdη. j=1 the values of q (j are kow at iteger times. (4.1 As explaied above, whe multiple causes of death are cosidered, icreasigess may become urealistic for some specific decremets. This is why we cosider death rates with various shapes i the ext result, all complyig with assumptio (4.1. Propositio 4.1. (i If µ 0j is either icreasig or decreasig the { } mi{q 1, (j q +1} (j q (j 1 q (j +1 µ 0j (t max,, t q 1 1 q +1 (ii If µ 0j is covex the ( { } q (j q (j 1 q (j +1 max, q(j µ 0j (t (4.2 1 q 1 1 q +1 1 q { } q (j 1 q (j +1 max,, t [, + 1]. 1 q 1 1 q +1 7
11 (iii If µ 0j is cocave the mi{q (j 1, q (j +1} µ 0j (t ( q(j + q (j mi{q (j 1 q 1, q +1} (j, t [, + 1]. Proof. Let us establish (i. Sice the itesities µ 0j (u are oegative, we have (4.3 ad hece +1 µ 0j (u (1 q du q (j +1 µ 0j (u du, q (j +1 µ 0j (u du q(j, (4.4 1 q whece the aouced iequality follows. Let us ow tur to (ii. As i the alivedead model, covexity implies that we first have decreasigess ad the icreasigess. Hece, { } q (j 1 q (j +1 µ 0j (t max,, t q 1 1 q +1 Also similar to the alivedead model, we have that if µ 0j (t q(j 1 q + c o t [, + 1], the µ 0j (t q (j c for all t [, + 1], which leads to the lower boud i (4.2. The proof for (iii is similar. Let N be the set of all trasitio itesities µ 0j, j = 1,..., m, that satisfy (4.1. Furthermore, we write l 0j (t ad u 0j (t for the lower ad upper boud o µ 0j (t at time t (accordig to Propositio 4.1(i, (ii or (iii. By applyig the worstcase method of Christiase (2010, we obtai the followig geeralizatio of Propositio 3.4. Propositio 4.2. The bestcase reserve V (t = if µ N V (t; µ ad worstcase reserves V (t = sup µ N V (t; µ uiquely solve the itegral equatios V (t =B 0 (ω x B 0 (t (t,ω x] V (s ϕ(s ds + 1 ( (b 0j (s V (s (u 0j (s + l 0j (s b 0j (s V (s (u 0j (s l 0j (s dt, 2 j=1 (t,ω x] V (t =B 0 (ω x B 0 (t V (s ϕ(s ds j=1 (t,ω x] (t,ω x] ( (b 0j (s V (s (u 0j (s + l 0j (s + b 0j (s V (s (u 0j (s l 0j (s dt. 8
12 mootoy assumptio reserve at age 30 reserve at age 50 upper boud true value lower boud covexity assumptio reserve at age 30 reserve at age 50 upper boud true value lower boud Table 2: Worstcase ad bestcase scearios for the combiatio of critical illess cover with edowmet isurace described i Example 4.4. For the worstcase reserve the proof is give i Christiase (2010. The proof for the bestcase reserve is aalogous. From the bestcase ad worstcase reserves we ca derive the correspodig trasitio itesities, as stated ext. Propositio 4.3. Defie µ 0j (t = 1 b0j (t V (tl 0j (t + 1 b0j (t<v (tu 0j (t, µ 0j (t = 1 b0j (t V (t u 0j(t + 1 b0j (t<v (t l 0j(t. The, V (t = V (t; µ ad V (t = V (t; µ for all t. The followig umerical example illustrates the accuracy of the bouds derived i Propositio 4.2. Example 4.4. Cosider a mixture of a edowmet isurace ad a critical illess isurace. If the policyholder gets a critical illess before age 65, a paymet of 2 is made ad the cotract termiates. If the policyholder does ot icur a critical illess but dies before age 65, a death beefit of 1 is paid. If oe of these two evets occurs, a survival beefit of 2 is made at age 65. Oly the trasitios active to dead ad active to ill are relevat here. We assume that the mortality itesity has the same form as i Example 3.6 ad that the morbidity itesity has the form µ 02 (t = exp(0.065t, but that the isurer has oly iformatio o q (1 ad q (2 at iteger times (see (4.1. Table 2 shows lower ad upper bouds for the prospective reserve i state active. We see that the bouds derived from Propositio 4.2 are accurate eough for practical purposes. Ackowledgemets Michel Deuit ackowledges the fiacial support of AG Isurace uder the K.U.Leuve Health Isurace Chair. 9
13 Refereces Barz, C., Müller, A., Compariso ad bouds for fuctioals of future lifetimes cosistet with life tables. Isurace: Mathematics ad Ecoomics 50, Christiase, M.C., Biometric worstcase scearios for multistate life isurace policies. Isurace: Mathematics ad Ecoomics 47, Christiase, M.C., Deuit, M.M., Firstorder mortality rates ad safeside actuarial calculatios i life isurace. ASTIN Bulleti 40, Frostig, E., Compariso betwee future lifetime distributio ad its approximatio. North America Actuarial Joural 6(2, Frostig, E., Properties of the power family of fractioal age approximatios. Isurace:Mathematics ad Ecoomics 33, Hossai, S.A., Quadratic fractioal age assumptio revisited. Lifetime Data Aal 17, Jemal, A., Ward, E., Hao, Y., Thu, M., Treds i the leadig causes of death i the Uited States, Joural of the America Medical Associatio 294, Joes B.L., Mereu J.A., A family of fractioal age assumptios. Isurace:Mathematics ad Ecoomics 27, Joes B.L., Mereu J.A., A critique of fractioal age assumptios. Isurace:Mathematics ad Ecoomics 30,
Subject CT5 Contingencies Core Technical Syllabus
Subject CT5 Cotigecies Core Techical Syllabus for the 2015 exams 1 Jue 2014 Aim The aim of the Cotigecies subject is to provide a groudig i the mathematical techiques which ca be used to model ad value
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationSequences and Series
CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their
More informationProperties of MLE: consistency, asymptotic normality. Fisher information.
Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS200609 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationA probabilistic proof of a binomial identity
A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two
More informationIrreducible polynomials with consecutive zero coefficients
Irreducible polyomials with cosecutive zero coefficiets Theodoulos Garefalakis Departmet of Mathematics, Uiversity of Crete, 71409 Heraklio, Greece Abstract Let q be a prime power. We cosider the problem
More informationInfinite Sequences and Series
CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...
More informationI. Chisquared Distributions
1 M 358K Supplemet to Chapter 23: CHISQUARED DISTRIBUTIONS, TDISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad tdistributios, we first eed to look at aother family of distributios, the chisquared distributios.
More informationWeek 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable
Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5
More informationTO: Users of the ACTEX Review Seminar on DVD for SOA Exam MLC
TO: Users of the ACTEX Review Semiar o DVD for SOA Eam MLC FROM: Richard L. (Dick) Lodo, FSA Dear Studets, Thak you for purchasig the DVD recordig of the ACTEX Review Semiar for SOA Eam M, Life Cotigecies
More informationPROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUSMALUS SYSTEM
PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUSMALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics
More informationInstitute of Actuaries of India Subject CT1 Financial Mathematics
Istitute of Actuaries of Idia Subject CT1 Fiacial Mathematics For 2014 Examiatios Subject CT1 Fiacial Mathematics Core Techical Aim The aim of the Fiacial Mathematics subject is to provide a groudig i
More informationDiscrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13
EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may
More information5: Introduction to Estimation
5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample
More informationConvexity, Inequalities, and Norms
Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for
More informationThe following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles
The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More information5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?
5.4 Amortizatio Questio 1: How do you fid the preset value of a auity? Questio 2: How is a loa amortized? Questio 3: How do you make a amortizatio table? Oe of the most commo fiacial istrumets a perso
More informationNonlife insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring
Nolife isurace mathematics Nils F. Haavardsso, Uiversity of Oslo ad DNB Skadeforsikrig Mai issues so far Why does isurace work? How is risk premium defied ad why is it importat? How ca claim frequecy
More informationChapter 7 Methods of Finding Estimators
Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of
More informationModel points and TailVaR in life insurance
Model poits ad TailVaR i life isurace Michel Deuit Istitute of Statistics, Biostatistics ad Actuarial Sciece Uiversité Catholique de Louvai (UCL) LouvailaNeuve, Belgium Julie Trufi Departmet of Mathematics
More informationConfidence Intervals for One Mean
Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a
More informationModified Line Search Method for Global Optimization
Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o
More informationVladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT
Keywords: project maagemet, resource allocatio, etwork plaig Vladimir N Burkov, Dmitri A Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT The paper deals with the problems of resource allocatio betwee
More informationSAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx
SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval
More informationThe Gompertz Makeham coupling as a Dynamic Life Table. Abraham Zaks. Technion I.I.T. Haifa ISRAEL. Abstract
The Gompertz Makeham couplig as a Dyamic Life Table By Abraham Zaks Techio I.I.T. Haifa ISRAEL Departmet of Mathematics, Techio  Israel Istitute of Techology, 32000, Haifa, Israel Abstract A very famous
More information.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth
Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,
More informationBENEFITCOST ANALYSIS Financial and Economic Appraisal using Spreadsheets
BENEITCST ANALYSIS iacial ad Ecoomic Appraisal usig Spreadsheets Ch. 2: Ivestmet Appraisal  Priciples Harry Campbell & Richard Brow School of Ecoomics The Uiversity of Queeslad Review of basic cocepts
More information1 Correlation and Regression Analysis
1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio
More informationSwaps: Constant maturity swaps (CMS) and constant maturity. Treasury (CMT) swaps
Swaps: Costat maturity swaps (CMS) ad costat maturity reasury (CM) swaps A Costat Maturity Swap (CMS) swap is a swap where oe of the legs pays (respectively receives) a swap rate of a fixed maturity, while
More informationAnnuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.
Auities Uder Radom Rates of Iterest II By Abraham Zas Techio I.I.T. Haifa ISRAEL ad Haifa Uiversity Haifa ISRAEL Departmet of Mathematics, Techio  Israel Istitute of Techology, 3000, Haifa, Israel I memory
More informationEkkehart Schlicht: Economic Surplus and Derived Demand
Ekkehart Schlicht: Ecoomic Surplus ad Derived Demad Muich Discussio Paper No. 200617 Departmet of Ecoomics Uiversity of Muich Volkswirtschaftliche Fakultät LudwigMaximiliasUiversität Müche Olie at http://epub.ub.uimueche.de/940/
More informationNEW HIGH PERFORMANCE COMPUTATIONAL METHODS FOR MORTGAGES AND ANNUITIES. Yuri Shestopaloff,
NEW HIGH PERFORMNCE COMPUTTIONL METHODS FOR MORTGGES ND NNUITIES Yuri Shestopaloff, Geerally, mortgage ad auity equatios do ot have aalytical solutios for ukow iterest rate, which has to be foud usig umerical
More informationCS103X: Discrete Structures Homework 4 Solutions
CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible sixfigure salaries i whole dollar amouts are there that cotai at least
More information0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5
Sectio 13 KolmogorovSmirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.
More informationChapter 6: Variance, the law of large numbers and the MonteCarlo method
Chapter 6: Variace, the law of large umbers ad the MoteCarlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value
More informationUC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 9 Spring 2006
Exam format UC Bereley Departmet of Electrical Egieerig ad Computer Sciece EE 6: Probablity ad Radom Processes Solutios 9 Sprig 006 The secod midterm will be held o Wedesday May 7; CHECK the fial exam
More informationTaking DCOP to the Real World: Efficient Complete Solutions for Distributed MultiEvent Scheduling
Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed MultiEvet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria
More informationEstimating Probability Distributions by Observing Betting Practices
5th Iteratioal Symposium o Imprecise Probability: Theories ad Applicatios, Prague, Czech Republic, 007 Estimatig Probability Distributios by Observig Bettig Practices Dr C Lych Natioal Uiversity of Irelad,
More informationSection 11.3: The Integral Test
Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult
More informationOutput Analysis (2, Chapters 10 &11 Law)
B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should
More informationHypothesis testing. Null and alternative hypotheses
Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate
More informationChapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions
Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig
More informationAmendments to employer debt Regulations
March 2008 Pesios Legal Alert Amedmets to employer debt Regulatios The Govermet has at last issued Regulatios which will amed the law as to employer debts uder s75 Pesios Act 1995. The amedig Regulatios
More informationCHAPTER 7: Central Limit Theorem: CLT for Averages (Means)
CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) X = the umber obtaied whe rollig oe six sided die oce. If we roll a six sided die oce, the mea of the probability distributio is X P(X = x) Simulatio:
More informationLecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)
18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the BruMikowski iequality for boxes. Today we ll go over the
More informationTrading the randomness  Designing an optimal trading strategy under a drifted random walk price model
Tradig the radomess  Desigig a optimal tradig strategy uder a drifted radom walk price model Yuao Wu Math 20 Project Paper Professor Zachary Hamaker Abstract: I this paper the author iteds to explore
More informationCHAPTER 3 DIGITAL CODING OF SIGNALS
CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity
More informationDecomposition of Gini and the generalized entropy inequality measures. Abstract
Decompositio of Gii ad the geeralized etropy iequality measures Stéphae Mussard LAMETA Uiversity of Motpellier I Fraçoise Seyte LAMETA Uiversity of Motpellier I Michel Terraza LAMETA Uiversity of Motpellier
More informationMARTINGALES AND A BASIC APPLICATION
MARTINGALES AND A BASIC APPLICATION TURNER SMITH Abstract. This paper will develop the measuretheoretic approach to probability i order to preset the defiitio of martigales. From there we will apply this
More informationTHE ABRACADABRA PROBLEM
THE ABRACADABRA PROBLEM FRANCESCO CARAVENNA Abstract. We preset a detailed solutio of Exercise E0.6 i [Wil9]: i a radom sequece of letters, draw idepedetly ad uiformly from the Eglish alphabet, the expected
More information4.3. The Integral and Comparison Tests
4.3. THE INTEGRAL AND COMPARISON TESTS 9 4.3. The Itegral ad Compariso Tests 4.3.. The Itegral Test. Suppose f is a cotiuous, positive, decreasig fuctio o [, ), ad let a = f(). The the covergece or divergece
More informationOur aim is to show that under reasonable assumptions a given 2πperiodic function f can be represented as convergent series
8 Fourier Series Our aim is to show that uder reasoable assumptios a give periodic fuctio f ca be represeted as coverget series f(x) = a + (a cos x + b si x). (8.) By defiitio, the covergece of the series
More informationA Recursive Formula for Moments of a Binomial Distribution
A Recursive Formula for Momets of a Biomial Distributio Árpád Béyi beyi@mathumassedu, Uiversity of Massachusetts, Amherst, MA 01003 ad Saverio M Maago smmaago@psavymil Naval Postgraduate School, Moterey,
More informationLecture 4: Cauchy sequences, BolzanoWeierstrass, and the Squeeze theorem
Lecture 4: Cauchy sequeces, BolzaoWeierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits
More informationActuarial Models for Valuation of Critical Illness Insurance Products
INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 9, 015 Actuarial Models for Valuatio of Critical Illess Isurace Products P. Jidrová, V. Pacáková Abstract Critical illess
More informationZTEST / ZSTATISTIC: used to test hypotheses about. µ when the population standard deviation is unknown
ZTEST / ZSTATISTIC: used to test hypotheses about µ whe the populatio stadard deviatio is kow ad populatio distributio is ormal or sample size is large TTEST / TSTATISTIC: used to test hypotheses about
More informationTHE HEIGHT OF qbinary SEARCH TREES
THE HEIGHT OF qbinary SEARCH TREES MICHAEL DRMOTA AND HELMUT PRODINGER Abstract. q biary search trees are obtaied from words, equipped with the geometric distributio istead of permutatios. The average
More informationThe analysis of the Cournot oligopoly model considering the subjective motive in the strategy selection
The aalysis of the Courot oligopoly model cosiderig the subjective motive i the strategy selectio Shigehito Furuyama Teruhisa Nakai Departmet of Systems Maagemet Egieerig Faculty of Egieerig Kasai Uiversity
More informationFIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix
FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. Powers of a matrix We begi with a propositio which illustrates the usefuless of the diagoalizatio. Recall that a square matrix A is diogaalizable if
More informationEntropy of bicapacities
Etropy of bicapacities Iva Kojadiovic LINA CNRS FRE 2729 Site école polytechique de l uiv. de Nates Rue Christia Pauc 44306 Nates, Frace iva.kojadiovic@uivates.fr JeaLuc Marichal Applied Mathematics
More informationInstallment Joint Life Insurance Actuarial Models with the Stochastic Interest Rate
Iteratioal Coferece o Maagemet Sciece ad Maagemet Iovatio (MSMI 4) Istallmet Joit Life Isurace ctuarial Models with the Stochastic Iterest Rate NiaNia JI a,*, Yue LI, DogHui WNG College of Sciece, Harbi
More information1. C. The formula for the confidence interval for a population mean is: x t, which was
s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : pvalue
More informationCenter, Spread, and Shape in Inference: Claims, Caveats, and Insights
Ceter, Spread, ad Shape i Iferece: Claims, Caveats, ad Isights Dr. Nacy Pfeig (Uiversity of Pittsburgh) AMATYC November 2008 Prelimiary Activities 1. I would like to produce a iterval estimate for the
More information5 Boolean Decision Trees (February 11)
5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected
More informationBOUNDS FOR THE PRICE OF A EUROPEANSTYLE ASIAN OPTION IN A BINARY TREE MODEL
BOUNDS FOR THE PRICE OF A EUROPEANSTYLE ASIAN OPTION IN A BINARY TREE MODEL HUGUETTE REYNAERTS, MICHELE VANMAELE, JAN DHAENE ad GRISELDA DEELSTRA,1 Departmet of Applied Mathematics ad Computer Sciece,
More informationYour organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:
Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network
More informationDetermining the sample size
Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS
More informationINFINITE SERIES KEITH CONRAD
INFINITE SERIES KEITH CONRAD. Itroductio The two basic cocepts of calculus, differetiatio ad itegratio, are defied i terms of limits (Newto quotiets ad Riema sums). I additio to these is a third fudametal
More informationLecture 2: Karger s Min Cut Algorithm
priceto uiv. F 3 cos 5: Advaced Algorithm Desig Lecture : Karger s Mi Cut Algorithm Lecturer: Sajeev Arora Scribe:Sajeev Today s topic is simple but gorgeous: Karger s mi cut algorithm ad its extesio.
More informationCS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations
CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationMaximum Likelihood Estimators.
Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio
More information*The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature.
Itegrated Productio ad Ivetory Cotrol System MRP ad MRP II Framework of Maufacturig System Ivetory cotrol, productio schedulig, capacity plaig ad fiacial ad busiess decisios i a productio system are iterrelated.
More informationNormal Distribution.
Normal Distributio www.icrf.l Normal distributio I probability theory, the ormal or Gaussia distributio, is a cotiuous probability distributio that is ofte used as a first approimatio to describe realvalued
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationAP Calculus AB 2006 Scoring Guidelines Form B
AP Calculus AB 6 Scorig Guidelies Form B The College Board: Coectig Studets to College Success The College Board is a otforprofit membership associatio whose missio is to coect studets to college success
More information3 Basic Definitions of Probability Theory
3 Basic Defiitios of Probability Theory 3defprob.tex: Feb 10, 2003 Classical probability Frequecy probability axiomatic probability Historical developemet: Classical Frequecy Axiomatic The Axiomatic defiitio
More informationFrance caters to innovative companies and offers the best research tax credit in Europe
1/5 The Frech Govermet has three objectives : > improve Frace s fiscal competitiveess > cosolidate R&D activities > make Frace a attractive coutry for iovatio Tax icetives have become a key elemet of public
More informationChapter 7  Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas:
Chapter 7  Samplig Distributios 1 Itroductio What is statistics? It cosist of three major areas: Data Collectio: samplig plas ad experimetal desigs Descriptive Statistics: umerical ad graphical summaries
More informationCHAPTER 3 THE TIME VALUE OF MONEY
CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all
More informationChapter 14 Nonparametric Statistics
Chapter 14 Noparametric Statistics A.K.A. distributiofree statistics! Does ot deped o the populatio fittig ay particular type of distributio (e.g, ormal). Sice these methods make fewer assumptios, they
More informationThe Stable Marriage Problem
The Stable Marriage Problem William Hut Lae Departmet of Computer Sciece ad Electrical Egieerig, West Virgiia Uiversity, Morgatow, WV William.Hut@mail.wvu.edu 1 Itroductio Imagie you are a matchmaker,
More informationwhere: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return
EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The
More informationINVESTMENT PERFORMANCE COUNCIL (IPC)
INVESTMENT PEFOMANCE COUNCIL (IPC) INVITATION TO COMMENT: Global Ivestmet Performace Stadards (GIPS ) Guidace Statemet o Calculatio Methodology The Associatio for Ivestmet Maagemet ad esearch (AIM) seeks
More informationApproximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find
1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.
More informationTHIN SEQUENCES AND THE GRAM MATRIX PAMELA GORKIN, JOHN E. MCCARTHY, SANDRA POTT, AND BRETT D. WICK
THIN SEQUENCES AND THE GRAM MATRIX PAMELA GORKIN, JOHN E MCCARTHY, SANDRA POTT, AND BRETT D WICK Abstract We provide a ew proof of Volberg s Theorem characterizig thi iterpolatig sequeces as those for
More informationNr. 2. Interpolation of Discount Factors. Heinz Cremers Willi Schwarz. Mai 1996
Nr 2 Iterpolatio of Discout Factors Heiz Cremers Willi Schwarz Mai 1996 Autore: Herausgeber: Prof Dr Heiz Cremers Quatitative Methode ud Spezielle Bakbetriebslehre Hochschule für Bakwirtschaft Dr Willi
More informationCooleyTukey. Tukey FFT Algorithms. FFT Algorithms. Cooley
Cooley CooleyTuey Tuey FFT Algorithms FFT Algorithms Cosider a legth sequece x[ with a poit DFT X[ where Represet the idices ad as +, +, Cooley CooleyTuey Tuey FFT Algorithms FFT Algorithms Usig these
More informationLesson 17 Pearson s Correlation Coefficient
Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) types of data scatter plots measure of directio measure of stregth Computatio covariatio of X ad Y uique variatio i X ad Y measurig
More information3. Greatest Common Divisor  Least Common Multiple
3 Greatest Commo Divisor  Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd
More informationCase Study. Normal and t Distributions. Density Plot. Normal Distributions
Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca
More informationStatistical inference: example 1. Inferential Statistics
Statistical iferece: example 1 Iferetial Statistics POPULATION SAMPLE A clothig store chai regularly buys from a supplier large quatities of a certai piece of clothig. Each item ca be classified either
More informationOverview of some probability distributions.
Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability
More informationData Analysis and Statistical Behaviors of Stock Market Fluctuations
44 JOURNAL OF COMPUTERS, VOL. 3, NO. 0, OCTOBER 2008 Data Aalysis ad Statistical Behaviors of Stock Market Fluctuatios Ju Wag Departmet of Mathematics, Beijig Jiaotog Uiversity, Beijig 00044, Chia Email:
More information1 Computing the Standard Deviation of Sample Means
Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.
More informationIs there employment discrimination against the disabled? Melanie K Jones i. University of Wales, Swansea
Is there employmet discrimiatio agaist the disabled? Melaie K Joes i Uiversity of Wales, Swasea Abstract Whilst cotrollig for uobserved productivity differeces, the gap i employmet probabilities betwee
More informationTheorems About Power Series
Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real oegative umber R, called the radius
More information