Distributions for Uncertainty Analysis 1

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Distributions for Uncertainty Analysis 1"

Transcription

1 Distributions for Uncertinty Anlysis 1 Howrd Cstrup, Ph.D. President, Integrted Sciences Group Bkersfield, CA Abstrct In performing mesurement, we encounter errors or bises from number of sources. Such sources include rndom error, mesuring prmeter bis, mesuring prmeter resolution, opertor bis, environmentl fctors, etc. We estimte the uncertinties due to these errors either by computing stndrd devition from smple of mesurements or by forming n estimte bsed on eperience. Estimtes obtined by the former method re lbeled Type A estimtes nd those obtined by the ltter method re clled Type B estimtes. This pper describes sttisticl distributions tht cn be pplied to both Type A nd Type B mesurement errors nd to equipment prmeter bises. Once the sttisticl distribution for mesurement error or bis is chrcterized, the uncertinty in this error or bis is computed s the stndrd devition of the distribution. For Type A estimtes, the distribution or popultion stndrd devition is estimted by the smple stndrd devition. For Type B estimtes, the stndrd devition is computed from limits, referred to s error continment limits nd from probbilities, referred to s continment probbilities. The degrees of freedom for ech uncertinty estimte cn often be determined, regrdless of whether the estimte is Type A or Type B. Bckground Until the publiction of the Guide to the Epression of Uncertinty in Mesurement (GUM) [1], ccrediting bodies or uditing gencies for test nd clibrtion orgniztions did not tend to focus on uncertinty nlysis requirements. There were two min resons for this: (1) universlly ccepted methodology ws not vilble, nd () ssessors nd uditors did not possess the required epertise. Since the introduction of the GUM, however, ccrediting bodies hve been incresingly insistent tht lbortories implement procedures for uncertinty nlysis nd be ble to demonstrte tht these procedures re being competently followed. Since the publiction of ISO/IEC 1705 [], this insistence hs intensified. This hs plced ccrediting bodies nd lbortories like in ctchup mode tht hs led to some hstily contrived mesures, s will be discussed presently. To induce orgniztions to estimte uncertinties, it ws felt necessry by some to dvocte the use of simple lgorithms tht, while they were not pproprite in most cses, would t lest get people on the uncertinty nlysis pth. One such lgorithm involves the indiscriminte use of the uniform distribution to compute Type B uncertinty estimtes. Unfortuntely, orgniztions tht not only wnt to nlyze uncertinties but lso do the job correctly re sometimes penlized by this ill-dvised simplifiction. On one occsion, lbortory ssessor dmitted tht the uniform distribution ws lrgely inpproprite but insisted tht it still be employed. His resoning ws tht it did not mtter if uncertinty estimtes were invlid s long s everyone produced them in the sme wy! This philosophy precludes the development of uncertinty estimtes tht cn be used to perform sttisticl tests, evlute mesurement decision risks, mnge clibrtion intervls, develop meningful tolernces nd compute vible confidence limits. In other words, prt from providing number, the uncertinty estimte becomes useless nd potentilly epensive commodity. Obviously, if vible uncertinty estimtes re to be produced, the blind cceptnce of inpproprite distributions is to be discourged. Accordingly, we need to elborte on lterntive distributions nd discuss the pplicbility of ech Introduction Error nd Uncertinty It is iomtic tht the uncertinty in vlue obtined by mesurement is identicl to the uncertinty in the mesurement error. Additionlly, the uncertinty in the vlue of tolernced prmeter or chrcterized 1 Presented t the 001 IDW Conference, Knoville, TN. Revised 7 My 004, to correct typogrphicl error in the cubic eqution for the qudrtic distribution. Revised 11 April 007 to provide more trctble form of the lognorml distribution.

2 reference stndrd is equl to the uncertinty in the prmeter s devition from its nominl or stted vlue. This iom cn be stted mthemticlly. The nottion is the following X - the true vlue of n ttribute - vlue obtined for the ttribute by mesurement or the ttribute s chrcterized or nominl vlue ε - the error in mesurement or devition from nominl or chrcterized vlue U - mthemticl opertor tht returns the uncertinty in vlue u - the uncertinty in u ε - the uncertinty in ε. We begin by sying tht Mesured Vlue = True Vlue + Mesurement Error, for mesured quntities, nd True Vlue = Nominl Vlue + Devition, for tolernced prmeters or chrcterized reference stndrds. We now rewrite these epressions using the nottion defined bove for mesured ttribute, nd = X + ε, (1) X = + ε, () for tolernced prmeter or chrcterized stndrd. Using the uncertinty opertor U, we obtin u = U( ) = U( X + ε) = U( ε) = u ε, (3) for mesured ttribute, nd ux = U( X) = U( + ε) = U( ε) = u ε, (4) for tolernced prmeter or chrcterized reference. In either cse, the uncertinty in the vlue of interest is equl to the uncertinty in the error or devition in the vlue. Uncertinty Definition We will now define the opertor U. First, however, we need to discuss the nture of mesurement errors nd devitions. We begin by stting tht mesurement errors nd devitions re rndom vribles tht follow sttisticl distributions. For certin kinds of error, such s rndom error, this is esily seen. For other kinds of error, such s prmeter bis nd opertor bis, however, their rndom nture is not so redily perceived. Wht we need to ber in mind is tht, while prticulr error my hve systemtic vlue tht persists from mesurement to mesurement, it nevertheless comes from some distribution of like errors tht cn be described sttisticlly. For instnce, the dimeters of bll berings emerging from mnufcturing process will vry to some finite mount from bering to bering. If one such bering comes into our possession, it will hve systemtic devition from nominl tht is essentilly fied. However, our prticulr devition ws drwn t rndom from popultion of devitions rising from the mnufcturing process. Since this devition is unknown, we cn tret it s rndom vrible whose uncertinty is mesure of the spred of devitions tht chrcterize the process. The wider this spred, the greter the uncertinty. A similr chin of resoning pplies to prmeters emerging from test or clibrtion process nd to errors in mesurement. The upshot is tht, whether prticulr error is rndom or systemtic, it cn still be regrded s coming from distribution of errors tht cn be described sttisticlly. Moreover, the spred in this distribution is synonymous with the uncertinty in the error. It turns out tht there is n idel sttistic for quntifying this spred. This sttistic is the stndrd devition of the distribution. Therefore, to define the opertor U, we need to define the stndrd devition. First, however, we will define the concept of sttisticl vrince. Simply put, the vrince of distribution of errors is the distribution s men squre error. If f() represents the probbility density for popultion of ttribute vlues or mesurement results, nd µ represents the nominl or men or vlue for the popultion, then the popultion vrince or men squre error vr(ε ) is given by u ε = vr( ε ) (5) = f( ε ) ε dε = f( )( µ ) d = vr( ) = u. Notice tht the popultion vrince is sttistic tht quntifies the spred of the distribution. Tht is, the lrger the spred, the lrger the vrince. At first glnce, the vrince or men squre error would seem to be good quntity by which to epress popul-

3 tion s uncertinty. However, the vrince is in the wrong units, nmely, the desired units squred. This is rectified by tking the squre root of the vrince, which yields the stndrd devition. Then, by Eq. (3) or (4) u = U( ) = U( ε ) = vr( ε ). (6) So, we see tht estimting the uncertinty in mesurement is n eercise in which we estimte the stndrd devition of the mesurement error. If we hve smple of mesurements, we cn estimte the stndrd devition due to rndom error in the smple using strightforwrd epression found in sttistics tetbooks 1 u = n ( i ), (7) n 1 i= 1 where n is the smple size nd is the smple men. This is n emple of Type A estimte. For Type B estimtes, we work from error continment limits nd continment probbilities. The process is described in detil in the literture [4]. Stndrd nd Epnded Uncertinty To this point, the uncertinty in mesurement hs been equted with the stndrd devition of the popultion of the mesurement error. In the GUM, this uncertinty is clled the stndrd uncertinty. If the distribution is known, nd the degrees of freedom cn be determined [4], the stndrd uncertinty cn be used to develop confidence limits for n uncertinty estimte. The GUM refers to confidence limit s n epnded uncertinty. 3 The fctor by which stndrd uncertinty is multiplied to yield n epnded uncertinty is clled the coverge fctor. Unfortuntely, in converstion, it is not lwys cler whether the term uncertinty refers to the epnded uncertinty or to the stndrd uncertinty. In this pper, unless otherwise indicted, it will refer to the stndrd uncertinty. Note the forml similrity between Eq. (7) nd Eq. (5). 3 Actully, the terms stndrd uncertinty nd epnded uncertinty were introduced to supersede the terms stndrd devition nd confidence limit, respectively, in cses where the degrees of freedom for n uncertinty estimte could not be determined. Before the refinement of methods for estimting degrees of freedom [4], this limittion pplied lmost universlly to Type B estimtes, nd, by etension to mied Type A-B estimtes. In this pper, uncertinty = stndrd uncertinty Sttisticl Distributions In obtining Type A uncertinty estimte, we compute stndrd devition using Eq. (7). In obtining Type B estimte, we work from set of bounding limits, referred to s error continment limits nd continment probbility, which is the probbility tht errors or ttribute vlues lie within these limits. Any one of vriety of distributions my be ssumed to represent the underlying distribution of errors or devitions. In this pper, we consider the uniform, norml, lognorml, qudrtic, cosine, hlf-cosine, U- shped, nd the Student s t distribution. The Uniform Distribution The uniform distribution is defined by the probbility density function (pdf) 1, f( ) = 0, otherwise, where ± re the limits of the distribution. f() 0 The Uniform Distribution. The probbility of lying between - nd is constnt. The probbility of lying outside ± is zero. Acceptnce of the Uniform Distribution Applying the uniform distribution to obtining Type B uncertinty estimtes is prctice tht hs been gining ground over the pst few yers. There re two min resons for this: 1. First, pplying the uniform distribution mkes it esy to obtin n uncertinty estimte. If the limits ± of the distribution re known, the uncertinty estimte is just u =. (8) 3

4 It should be sid tht the "ese of use" dvntge hs been promoted by individuls who re ignornt of methods of obtining uncertinty estimtes for more pproprite distributions nd by others who re simply looking for quick solution. In firness to the ltter group, they sometimes ssert tht the lck of specificity of informtion required to use other distributions mkes for crude uncertinty estimtes nywy, so why not get your crude estimte by intentionlly using n inpproprite distribution? At our present level of nlyticl development [3, 4], this rgument does not hold wter. Since the introduction of the GUM, methods hve been developed tht systemtize nd rigorize the use of distributions tht re physiclly relistic. These will be discussed presently.. Second, it hs been sserted by some tht the use of the uniform distribution is (uniformly?) recommended in the GUM. This is not true. In fct, most of the methodology of the GUM is bsed on the ssumption tht the underlying error distribution is norml. Some of the belief tht the uniform distribution is clled for in the GUM stems from the fct tht severl individuls, who hve come to be regrded s GUM uthorities, hve been dvocting its use. For clrifiction on this issue, the reder is referred to Section 4.3 of the GUM. Another source of confusion is tht some of the emples in the GUM pply the uniform distribution in situtions tht pper to be incomptible with its use. It is resonble to suppose tht much of this is due to the fct tht rigorous Type B estimtion methods nd tools were not vilble t the time the GUM ws published, nd the uniform distribution ws n "esy out." As stted in item 1 bove, the lck of such methods nd tools hs since been rectified. The cceptnce of the uniform distribution on the bsis of its use in GUM emples reminds us of similr prctice tht emerged from the ppliction of Hndbook 5 to the interprettion of MIL-STD- 4566A. In one emple in the Hndbook, hypotheticl lb ws being udited whose nominl operting temperture ws 68 F. Some of the 4566A uditors rected to the emple by citing lbs tht did not mintin this temperture, regrdless of whether it ws pproprite for the lb's opertion. Inevitbly, the 68 F requirement ctully becme institutionlized within certin uditing gencies. Applicbility of the Uniform Distribution The use of the uniform distribution is pproprite under limited set of conditions. These conditions re summrized by the following criteri. The first criterion is tht we must know set of minimum bounding limits for the distribution. This is the minimum limits criterion. Second, we must be ble to ssert tht the probbility of finding vlues between these limits is unity. This is the 100% continment criterion. Third, we must be ble to demonstrte tht the probbility of obtining vlues between the minimum bounding limits is uniform. This is the uniform probbility criterion. Minimum Limits Criterion. It is vitl tht the limits we estblish for the uniform distribution re the minimum bounding limits. For instnce, if the limits ±L bound the vrible of interest, then so do the limits ±L, ±3L, nd so on. Since the uncertinty estimte for the uniform distribution is obtined by dividing the bounding limit by the squre root of three, using vlue for the limit tht is not the minimum bounding vlue will obviously result in n invlid uncertinty estimte. This lone mkes the ppliction of the uniform distribution questionble in estimting bis uncertinty from such quntities s tolernce limits, for instnce. It my be tht out-of-tolernces hve never been observed for prticulr prmeter (100% continment), but it is unknown whether the tolernces re minimum bounding limits. Some yers go, study ws conducted involving voltge reference tht showed tht vlues for one prmeter were normlly distributed with stndrd devition tht ws pproimtely 1/10 of the tolernce limit. With 10-sigm limits, it is unlikely tht ny out-of-tolernces would be observed. However, if the uniform distribution were used to estimte the bis uncertinty for this item, bsed on tolernce limits, the uncertinty estimte would be nerly si times lrger thn would be pproprite. Some might clim tht this is cceptble, since the estimte cn be considered conservtive one. Tht my be. However, it is lso useless estimte. This point will be elborted lter. A second difficulty we fce when ttempting to pply minimum bounding limits is tht such limits cn rrely be estblished on physicl grounds. This is especilly true when using prmeter tolernce limits. It is virtully impossible to imgine sitution where design engineers hve somehow been ble to precisely identify the minimum limits tht bound vlues tht re physiclly ttinble. If we dd to this the fct tht tolernce limits re often influenced by mrketing

5 rther thn engineering considertions, equting tolernce limits with minimum bounding limits becomes very unfruitful nd misleding prctice. 100% Continment Criterion. By definition, the estblishment of minimum bounding limits implies the estblishment of 100% continment. It should be sid however, tht n uncertinty estimte my still be obtined for the uniform distribution if continment probbility less tht 100% is pplied. For instnce, suppose the continment limits re given s ±L nd the continment probbility is stted s being equl to some vlue p between zero nd one. Then, if the uniform probbility criterion is met, the limits of the distribution re given by L =, L. (9) p If the uniform probbility criterion is not met, however, the uniform distribution would not be pplicble, nd we should turn to other distributions. Uniform Probbility Criterion. As discussed bove, estblishing minimum continment limits cn be chllenging prospect. Hrder still is finding relworld mesurement error distributions tht demonstrte uniform probbility of occurrence between two limits nd zero probbility of occurrence outside these limits. Ecept in very limited instnces, such s re discussed in the net section, ssuming uniform probbility is just not physiclly relistic. This is true even in some cses where the distribution would pper to be pplicble. For emple, conjecture hs recently been dvnced tht the distribution of prmeters immeditely following test or clibrtion cn be sid to be uniform. While this seems resonble t fce vlue, it turns out not to be the cse. Becuse of flse ccept risk (consumer s risk), such distributions rnge from pproimtely tringulr to hving "humped" ppernce with rolled-off shoulders. As to whether we cn tret prmeter tolernce limits s bounds tht contin vlues with uniform probbility, we must imgine tht, not only hs the instrument mnufcturer mnged to mirculously scertin minimum bounding limits, but hs lso juggled physics to such n etent s to mke the prmeter vlue's probbility distribution uniform between these limits nd zero outside them. This would be truly mzing fet of engineering for most tolernced quntities especilly considering the mrketing influence mentioned erlier. Cses tht Stisfy the Criteri Digitl Resolution Uncertinty. We sometimes need to estimte the uncertinty due to the resolution of digitl redout. For instnce, three-digit redout might indicte V. If the device employs the stndrd round-off prctice, we know tht the displyed number is derived from sensed vlue tht lies between V nd V. We lso cn ssert to very high degree of vlidity tht the vlue hs n equl probbility of lying nywhere between these two numbers. In this cse, the use of the uniform distribution is pproprite, nd the resolution uncertinty is V u V = = V. 3 RF Phse Angle. RF power incident on lod my be delivered to the lod with phse ngle θ between -π nd π. In ddition, unless there is compelling reson to believe otherwise, the probbility of occurrence between these limits is uniform. Accordingly, the use of the uniform distribution is pproprite. This yields phse ngle uncertinty estimte of π u θ = It is interesting to note tht, given the bove, if we ssume tht the mplitude of the signl is sinusoidl, the distribution for incident voltge is the U-shped distribution. Quntiztion Error. The potentil drop (or lck of potentil drop) sensed cross ech element of n A/D Converter sensing network produces either "1" or "0" to the converter. This response constitutes "bit" in the binry code tht represents the smpled vlue. For ldder-type networks, the position of the bit in the code is determined by the loction of its originting network element. Even if no errors were present in smpling nd sensing the input signl, errors would still be introduced by the discrete nture of the encoding process. Suppose, for emple, tht the full scle signl level (dynmic rnge) of the A/D Converter is volts. If n bits re used in the encoding process, then voltge V cn be resolved into n discrete steps, ech of size / n. The error in the voltge V is thus ε ( V) = V m, where m is some integer determined by the sensing function of the D/A Converter. n

6 The continment limit ssocited with ech step is one-hlf the vlue of the mgnitude of the step. Consequently, the continment limit inherent in quntizing voltge V is (1/)(/ n ), or / n+1. This is embodied in the epression V V quntized = sensed ±. n+ 1 The uncertinty due to quntiztion error is obtined from the continment limits nd from the ssumption tht the sensed nlog vlue hs equl probbility of occurrence between these limits: u V / = 3 n+ 1. One wy of reconciling the prctice is to stte tht the underlying distribution is ctully norml, or pproimtely norml, nd the uniform distribution is used merely s n rtifice to obtin n estimte of the distribution's stndrd devition. This is somewht mzing sttement. If the underlying distribution is norml, why not obtin the uncertinty estimte using tht distribution in the first plce? 4 It cn be shown tht using the uniform distribution s tool for estimting the uncertinty in normlly distributed quntity corresponds to ssuming norml distribution with 91.67% continment probbility. For orgniztions tht mintin high in-tolernce probbility t the unit level, we often see or cn surmise 98% or better in-tolernce probbilities t the prmeter level. Consequently, for these cses, use of the uniform distribution produces uncertinty estimtes tht re t lest 35% lrger thn wht is pproprite. As for those who find this cceptble on the bsis of conservtism, consider the U.S. Nvy's end-of-period relibility trget of 7% for generl purpose items. For single-prmeter items, if the true underlying distribution is norml, use of the uniform distribution cn produce uncertinty estimtes tht re only bout 6% of wht they should be. So much for conservtism. Signl Quntiztion. The smpled signl points re quntized in multiples of discrete step size. Development of Epnded Uncertinty Limits NIST Technicl Note 197 [6] documents the uncertinty nlysis policy to be followed by NIST. In this policy, epnded uncertinty limits for Type B nd mied estimtes re obtined by multiplying the uncertinty estimte by fied k-fctor equl to two. Assuming n underlying norml distribution, this produces limits tht re roughly nlogous to 95% confidence limits. The dvisbility of this prctice is debtble, but this is the subject of seprte discussion. For the present, we consider wht results from the prctice when estimting n uncertinty for cse where the underlying distribution is ssumed to be uniform. Since the uncertinty is estimted by dividing the distribution minimum bounding limit by the squre root of three, multiplying this estimte by two yields epnded uncertinty limits tht re outside the distribution s minimum bounding limits. To be specific, these limits equte to pproimtely 115% continment probbility, which is nonsense. The Norml Distribution When obtining Type A estimte, we compute stndrd devition from smple of vlues. For emple, we estimte rndom uncertinty by computing 4 One recommendtion tht the reder my encounter is tht, if ll tht is vilble for n error source or prmeter devition is set of bounding limits, without ny knowledge of the nture of the error distribution nd with no informtion regrding continment probbility, then the uniform distribution should be ssumed. There re two points tht should be mde concerning this recommendtion. First, fter little reflection on the difficulty of obtining minimum continment limits without knowledge of continment probbility, we cn see tht the recommendtion not dvisble. The prudent pth to follow is to simply put some effort into obtining continment probbility estimte nd scertining most likely underlying distribution. There is relly no wy round this. Moreover, the uthor hs yet to observe n uncertinty nlysis problem where this could not be done. The second point is tht, eperienced technicl personnel nerly lwys know something bout wht they re mesuring nd wht they re mesuring it with. Ecept for the cses described bove, it is difficult to imgine scenrio where n eperienced engineer or technicin would know set of bounding limits nd nothing else.

7 the stndrd devition for smple of repeted mesurements of given vlue. We lso obtin smple size. The smple stndrd devition, equted with the rndom uncertinty of the smple, is n estimte of the stndrd devition for the popultion from which the smple ws drwn. Ecept in rre cses, we ssume tht this popultion follows the norml distribution. This ssumption, llows us to esily obtin the degrees of freedom nd the smple stndrd devition nd to construct confidence limits, perform sttisticl tests, estimte mesurement decision risk nd to rigorously combine the rndom uncertinty estimte with other Type A uncertinty estimtes. Why do we ssume norml distribution? The primry reson is becuse this is the distribution tht either represents or pproimtes wht we frequently see in the physicl universe. It cn be derived from the lws of physics for such phenomen s the diffusion of gses nd is pplicble to instrument prmeters subject to rndom stresses of usge nd hndling. It is lso often pplicble to equipment prmeters emerging from mnufcturing processes. f() In cses where this is not so, other distributions, such s the lognorml distribution cn be pplied. Uncertinty Estimtes In pplying the norml distribution, n uncertinty estimte is obtined from continment limits nd continment probbility. The use of the distribution is pproprite in cses where the bove considertions pply nd the limits nd probbility re t lest pproimtely known. The etent to which this knowledge is pproimte determines the degrees of freedom of the uncertinty estimte [4, 7]. The degrees of freedom nd the uncertinty estimte cn be used in conjunction with the Student's t distribution (see below) to compute confidence limits. Let ± represent the known continment limits nd let p represent the continment probbility. Then n estimte of the stndrd devition of the popultion of errors or devitions is obtined from u =, (10) 1 1+ p Φ where Φ -1 (. ) is the inverse norml distribution function. This function cn be found in sttistics tets nd in populr spredsheet progrms. µ The Norml Distribution. Shown is cse where the popultion men µ is locted fr from physicl limit 0. In such cses, the norml distribution cn be used without compromising rigor. An dditionl considertion pplies to the distribution we should ssume for totl error or devition tht is composed of constituent errors or devitions. There is theorem clled the centrl limit theorem tht demonstrtes tht, even though the individul constituent errors or devitions my not be normlly distributed, the combined error or devition is pproimtely so. An rgument hs been presented ginst the use of the norml distribution in cses where the vrible of interest is restricted, i.e., where vlues of the vrible re sid to be bound by some physicl limit. This condition notwithstnding, the norml distribution is still widely pplicble in tht, for mny such cses, the physicl limit is locted fr from the popultion men. If only single continment limit is pplicble, such s with single-sided tolernces, the pproprite epression is u =. (11) Φ p 1 ( ) The Lognorml Distribution The lognorml distribution cn often be used to estimte the uncertinty in equipment prmeter bis in cses where the tolernce limits re symmetric. It is lso used in cses where physicl limit is present tht lies close enough to the nominl or mode vlue to skew the prmeter bis pdf in such wy tht the norml distribution. is not pplicble. The pdf is given by 1 q f( ) = ep ln σ πσ q m q, where q is physicl limit for, m is the popultion medin nd µ is the popultion mode. The vrible σ is not the popultion stndrd devition. It is referred

8 to s the "shpe prmeter." The ccompnying grphic shows cse where µ = 10, q = 9.607, σ = , nd m = The computed stndrd devition for this emple is u = f() q µ The Lognorml Distribution. Useful for describing distributions for prmeters constrined by physicl limit or possessing symmetric tolernces. Uncertinty estimtes (stndrd devitions) for the lognorml distribution re obtined by numericl itertion. To dte, the only known pplictions tht perform this process re UncertintyAnlyzer [3] nd AccurcyRtio [5]. The Tringulr Distribution The tringulr distribution hs been proposed for use in cses where the continment probbility is 100%, but there is centrl tendency for vlues of the vrible of interest [1]. The tringulr distribution is the simplest distribution possible with these chrcteristics. f() - 0 The Tringulr Distribution. A distribution tht sometimes pplies to prmeter vlues immeditely following test or clibrtion. The pdf for the distribution is + ( )/, 0 f( ) = ( )/, 0 0, otherwise. The stndrd devition for the distribution is obtined from u =. (1) 6 Like the uniform distribution, using the tringulr distribution requires the estblishment of minimum continment limits ±. The sme reservtions pply in this regrd to the tringulr distribution s to the uniform distribution. In cses where continment probbility p < 1 cn be determined for limits ±L, where L <, the limits of the distribution re given by L =, L. 1 1 p Aprt from representing post-test distributions under certin restricted conditions, the tringulr distribution hs limited pplicbility to physicl errors or devitions. While it does not suffer from the uniform probbility criterion, s does the uniform distribution, it nevertheless displys brupt trnsitions t the bounding limits nd t the zero point, which re physiclly unrelistic in most instnces. In ddition, the liner increse nd decrese in behvior is somewht fnciful for pdf. The Qudrtic Distribution A distribution tht elimintes the brupt chnge t the zero point, does not ehibit unrelistic liner behvior nd stisfies the need for centrl tendency is the qudrtic distribution. This distribution is defined by the pdf 3 1 ( / ) f( ) = 4, 0, otherwise where ± re minimum bounding limits. The stndrd devition for this distribution is determined from u =, (13) 5 i.e., bout 77% of the stndrd devition estimte for the uniform distribution.

9 f() f() - 0 The Qudrtic Distribution. Ehibits centrl tendency without discontinuities nd does not ssume liner pdf behvior. For continment probbility p nd continment limits ±L, the minimum bounding limits ± re obtined from L 1 = p p p + < < 3 1 cos rccos(1 ) 1 1 The Cosine Distribution While the qudrtic distribution elimintes discontinuities within the bounding limits, it rises bruptly t the limits. Although the qudrtic distribution hs wider pplicbility thn either the tringulr or uniform distribution, this feture nevertheless diminishes its physicl vlidity. A distribution tht overcomes this shortcoming, ehibits centrl tendency nd cn be determined from minimum continment limits is the cosine distribution. The pdf for this distribution is given by 1 π 1 + cos f( ) =,. 0, otherwise The uncertinty is obtined from the epression 6 u = 1, (14) 3 π which trnsltes to roughly 63% of the vlue obtined using the uniform distribution The Cosine Distribution. A 100% continment distribution with centrl tendency nd lcking discontinuities. Solving for when continment probbility nd continment limits ±L re given requires pplying numericl itertive method to the epression 1 sin( π ) p + = 0, L/ ; L. π The solution lgorithm hs been implemented in the sme softwre lluded to in the discussion on the qudrtic distribution. It yields, for the ith itertion, where nd = F F, i i 1 / 1 F = sin( π ) p + π F = 1+ cos( π ). The Hlf-Cosine Distribution The hlf-cosine distribution is used in cses where the centrl tendency is not s pronounced s when norml or the cosine distribution would be pproprite. In this regrd, it resembles the qudrtic distribution without the discontinuities t the distribution limits. The pdf is π π cos f( ) = 4,. 0, otherwise If the minimum limiting vlues ± re known, the uncertinty is obtined from the epression u = 1 8/ π. (15) If continment limits ±L nd continment probbility p re known, the limiting vlues my be obtined from the reltion π L =, L. 1 sin ( p)

10 f() The Student's t Distribution If the underlying distribution is norml, nd Type A estimte nd degrees of freedom re vilble, confidence limits for mesurement errors or prmeter devitions my be obtined using the Student's t distribution. This distribution is vilble in sttistics tetbooks nd populr spredsheet pplictions. Its pdf is ν + 1 Γ f( ) = (1 + / ν ) ν πν Γ ( ν + 1)/, - 0 The Hlf-Cosine Distribution. Possesses centrl tendency but ehibits higher probbility of occurrence ner the minimum limiting vlues thn either the cosine or the norml distribution. The U Distribution The U distribution pplies to sinusoidl RF signls incident on lod. It hs the pdf 1, < < f( ) = π 0, otherwise, where represents the mimum signl mplitude. The uncertinty in the incident signl mplitude is estimted ccording to u =. (16) where ν is the degrees of freedom nd Γ(. ) is the gmm function. The degrees of freedom quntifies the mount of knowledge used in estimting uncertinty. This knowledge is incomplete if the limits ± re pproimte nd the continment probbility p is estimted from recollected eperience. Since the knowledge is incomplete, the degrees of freedom ssocited with Type B estimte is not infinite. If the degrees of freedom vrible is finite but unknown, the uncertinty estimte cnnot be rigorously used to develop confidence limits, perform sttisticl tests or mke decisions. This limittion hs often precluded the use of Type B estimtes s sttisticl quntities nd hs led to such discomforting rtifices s fied coverge fctors. f() f() 0-0 The U Distribution. The distribution is the pdf for sine wves of rndom phse incident on plne. If continment limits ±L nd continment probbility p re known, the prmeter cn be computed ccording to L = L. sin π p / ( ), Student's t Distribution. Shown is the pdf for 10 degrees of freedom. Fortuntely, the GUM provides n epression for obtining the pproimte degrees of freedom for Type B estimtes. However, the epression involves the use of the vrince in the uncertinty estimte, nd method for obtining this vrince hs been lcking until recently [4]. A rigorous method for obtining this quntity hs been implemented in commercilly vilble softwre [3] nd in freewre ppliction [7]. Once the degrees of freedom hs been obtined, the Type B estimte my then be combined with other estimtes nd the degrees of freedom for the combined

11 uncertinty cn be determined using the Welch- Stterthwite reltion [1]. If the underlying distribution for the combined estimte is norml, the t distribution cn be used to develop confidence limits nd perform sttisticl tests. The procedure is to first estimte the uncertinty using Eq. (10) nd then estimte the degrees of freedom from the epression where ( u) 1 1 σ ν B u 3ϕ ϕ ( ) + π e ( p) ϕ 1 1+ p ϕ =Φ., (17) The vribles nd p represent "give or tke" vlues for the continment limits nd continment probbility, respectively. At first glnce, Eq. (17) my seem to be nything but rigorous. However, severl dt input formts hve been developed tht rigorize the process of estimting nd p [4]. They re vilble in the referenced softwre pplictions cited bove [3, 7]. Striving for Conservtive Estimtes If n uncertinty estimte is viewed s n end product tht will be filed wy without ppliction of ny kind, then employing unrelistic distributions nd fied coverge fctors my be considered cceptble by some. Such distributions cn yield sttisticlly vlid estimtes, regrdless of whether or not these estimtes re physiclly vlid. However, if n uncertinty estimte is to be employed in mking decisions, such s my result from hypothesis testing or decision risk nlyses, employing physiclly unrelistic distribution is to be discourged. In these cses, dvocting the use of such distribution on the grounds tht it yields conservtive uncertinty estimtes is s irresponsible s employing intentionlly bised instruments to obtin mesurements tht re fvorbly skewed in one direction or nother. In ddition, the use of unrelistic distributions my yield estimtes tht re considerbly smller thn wht is pproprite under certin conditions. The emple of estimting bis uncertinty for singleprmeter Nvy generl purpose items, mentioned erlier, is cse in point. Another considertion tht rgues ginst employing conservtive uncertinty estimtes is tht this prctice sometimes leds to "reckless" conclusions. This is the cse when mesurement from one lbortory is tested ginst mesurement from nother to ssess equivlence between lbortories. If conservtive estimtes re used, the test ctully becomes less stringent thn otherwise. The bottom line is tht conservtive uncertinty estimtes re essentilly zero-informtion quntities tht hve no legitimte use. If conservtism is desired, it cn be implemented by insisting on high confidence levels in estimting confidence limits fter vlid uncertinty estimte is obtined. The higher the confidence level, the wider (more conservtive) the confidence limits. Recommendtions for Selecting Distributions Unless informtion to the contrry is vilble, the norml distribution should be pplied s the defult distribution. For Type B estimtes, the dt input formts lluded under the discussion of the Student's t distribution should lso be employed to estimte the degrees of freedom. If it is suspected tht the distribution of the vlue of interest is skewed, pply the lognorml distribution. In using the norml or lognorml distribution, some effort must be mde to estimte continment probbility. If set of continment limits is vilble, but 100% continment hs been observed, then the following is recommended: 1. If the vlue of interest hs been subjected to rndom usge or hndling stress, nd is ssumed to possess centrl tendency, pply the cosine distribution. If it is suspected tht vlues re more evenly distributed, pply either the qudrtic or hlf-cosine distribution, s pproprite. The tringulr distribution my be pplicble, under certin circumstnces, when deling with prmeters following testing or clibrtion.. If the vlue of interest is the mplitude of sine wve incident on plne with rndom phse, pply the U distribution. 3. If the vlue of interest is the resolution uncertinty of digitl redout, pply the uniform distribution. This distribution is lso pplicble to estimting the uncertinty due to quntiztion error nd the uncertinty in RF phse ngle.

12 Generl Procedure for Obtining Uncertinty Estimtes Type A Estimtes In mking Type A estimte nd using it to construct confidence limits, we pply the following procedure tken from the GUM nd elsewhere: 1. Tke rndom smple of size n representtive of the popultion of interest. The lrger the smple size, the better. In mny cses, smple size less thn si is not sufficient.. Compute smple stndrd devition, u using Eq. (7). 3. Assume n underlying distribution, e.g., norml. 4. Develop coverge fctor bsed on the degrees of freedom (n 1) ssocited with the smple stndrd devition nd desired level of confidence. If the underlying distribution is ssumed to be norml, use either t-tbles or Student s t spredsheet functions. In Microsoft Ecel, for emple, twosided coverge fctor cn be determined using the TINV function: t = TINV((1 p), ν ), where p is the confidence level nd ν is the degrees of freedom. 5. Multiply the smple stndrd devition by the coverge fctor to obtin L = tu nd use ±L s p 100% confidence limits. References [1] [] [3] [4] [5] [6] [7] ISO/TAG4/WG3, Guide to the Epression of Uncertinty in Mesurement, Interntionl Orgniztion for Stndrdiztion (ISO), Genev, ISO/IEC (E), Generl Requirements for the Competence of Testing nd Clibrtion Lbortories, ISO/IEC, December 15, UncertintyAnlyzer, , Integrted Sciences Group, All Rights Reserved. Cstrup, H., "Estimting Ctegory B Degrees of Freedom," Proc. Mesurement Science Conference, Jnury 000, Anheim. AccurcyRtio, , Integrted Sciences Group, All Rights Reserved. Tylor, B. nd Kuytt, C., NIST Technicl Note 197, "Guidelines for Evluting nd Epressing the Uncertinty of NIST Mesurement Results," U.S. Dept. of Commerce, ISG Ctegory B Uncertinty Clcultor, 000, Integrted Sciences Group, All Rights Reserved. Avilble from Type B Estimtes In mking Type B estimte, we reverse the process. The procedure is 1. Tke set of confidence limits, e.g., prmeter tolernce limits ±L (continment limits).. Estimte the confidence level, e.g., the in-tolernce probbility (continment probbility). 3. Estimte the degrees of freedom using Eq. (17). 4. Assume n underlying distribution, e.g., norml Compute coverge fctor, t, bsed on the continment probbility nd degrees of freedom. 6. Compute the stndrd uncertinty for the quntity of interest (e.g., prmeter bis) by dividing the confidence limit by the coverge fctor: u = L/ t. 5 The Type B estimtion procedure hs been refined so tht stndrd devitions cn be estimted for non-norml popultions nd in cses where the confidence limits re symmetric or even single-sided [3, 5].

9 CONTINUOUS DISTRIBUTIONS

9 CONTINUOUS DISTRIBUTIONS 9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete

More information

Tests for One Poisson Mean

Tests for One Poisson Mean Chpter 412 Tests for One Poisson Men Introduction The Poisson probbility lw gives the probbility distribution of the number of events occurring in specified intervl of time or spce. The Poisson distribution

More information

STA 2023 Test #3 Practice Multiple Choice

STA 2023 Test #3 Practice Multiple Choice STA 223 Test #3 Prctice Multiple Choice 1. A newspper conducted sttewide survey concerning the 1998 rce for stte sentor. The newspper took rndom smple (ssume it is n SRS) of 12 registered voters nd found

More information

Treatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.

Treatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3. The nlysis of vrince (ANOVA) Although the t-test is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the t-test cn be used to compre the mens of only

More information

Distributions. (corresponding to the cumulative distribution function for the discrete case).

Distributions. (corresponding to the cumulative distribution function for the discrete case). Distributions Recll tht n integrble function f : R [,] such tht R f()d = is clled probbility density function (pdf). The distribution function for the pdf is given by F() = (corresponding to the cumultive

More information

Operations with Polynomials

Operations with Polynomials 38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply

More information

Lecture 3 Gaussian Probability Distribution

Lecture 3 Gaussian Probability Distribution Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike

More information

NOTES AND CORRESPONDENCE. Uncertainties of Derived Dewpoint Temperature and Relative Humidity

NOTES AND CORRESPONDENCE. Uncertainties of Derived Dewpoint Temperature and Relative Humidity MAY 4 NOTES AND CORRESPONDENCE 81 NOTES AND CORRESPONDENCE Uncertinties of Derived Dewpoint Temperture nd Reltive Humidity X. LIN AND K. G. HUBBARD High Plins Regionl Climte Center, School of Nturl Resource

More information

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive

More information

Chapter 6 Solving equations

Chapter 6 Solving equations Chpter 6 Solving equtions Defining n eqution 6.1 Up to now we hve looked minly t epressions. An epression is n incomplete sttement nd hs no equl sign. Now we wnt to look t equtions. An eqution hs n = sign

More information

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( ) Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +

More information

Solutions to Section 1

Solutions to Section 1 Solutions to Section Exercise. Show tht nd. This follows from the fct tht mx{, } nd mx{, } Exercise. Show tht = { if 0 if < 0 Tht is, the bsolute vlue function is piecewise defined function. Grph this

More information

Or more simply put, when adding or subtracting quantities, their uncertainties add.

Or more simply put, when adding or subtracting quantities, their uncertainties add. Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re

More information

Ae2 Mathematics : Fourier Series

Ae2 Mathematics : Fourier Series Ae Mthemtics : Fourier Series J. D. Gibbon (Professor J. D Gibbon, Dept of Mthemtics j.d.gibbon@ic.c.uk http://www.imperil.c.uk/ jdg These notes re not identicl word-for-word with my lectures which will

More information

Integration. 148 Chapter 7 Integration

Integration. 148 Chapter 7 Integration 48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but

More information

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions. Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd

More information

1 Numerical Solution to Quadratic Equations

1 Numerical Solution to Quadratic Equations cs42: introduction to numericl nlysis 09/4/0 Lecture 2: Introduction Prt II nd Solving Equtions Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mrk Cowlishw Numericl Solution to Qudrtic Equtions Recll

More information

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered: Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you

More information

Experiment 6: Friction

Experiment 6: Friction Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht

More information

Section A-4 Rational Expressions: Basic Operations

Section A-4 Rational Expressions: Basic Operations A- Appendi A A BASIC ALGEBRA REVIEW 7. Construction. A rectngulr open-topped bo is to be constructed out of 9- by 6-inch sheets of thin crdbord by cutting -inch squres out of ech corner nd bending the

More information

Physics 43 Homework Set 9 Chapter 40 Key

Physics 43 Homework Set 9 Chapter 40 Key Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nm-wide region t x

More information

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of

More information

Quadratic Equations. Math 99 N1 Chapter 8

Quadratic Equations. Math 99 N1 Chapter 8 Qudrtic Equtions Mth 99 N1 Chpter 8 1 Introduction A qudrtic eqution is n eqution where the unknown ppers rised to the second power t most. In other words, it looks for the vlues of x such tht second degree

More information

14.2. The Mean Value and the Root-Mean-Square Value. Introduction. Prerequisites. Learning Outcomes

14.2. The Mean Value and the Root-Mean-Square Value. Introduction. Prerequisites. Learning Outcomes he Men Vlue nd the Root-Men-Squre Vlue 4. Introduction Currents nd voltges often vry with time nd engineers my wish to know the men vlue of such current or voltge over some prticulr time intervl. he men

More information

Uniform convergence and its consequences

Uniform convergence and its consequences Uniform convergence nd its consequences The following issue is centrl in mthemtics: On some domin D, we hve sequence of functions {f n }. This mens tht we relly hve n uncountble set of ordinry sequences,

More information

Matrix Algebra CHAPTER 1 PREAMBLE 1.1 MATRIX ALGEBRA

Matrix Algebra CHAPTER 1 PREAMBLE 1.1 MATRIX ALGEBRA CHAPTER 1 Mtrix Algebr PREAMBLE Tody, the importnce of mtrix lgebr is of utmost importnce in the field of physics nd engineering in more thn one wy, wheres before 1925, the mtrices were rrely used by the

More information

All pay auctions with certain and uncertain prizes a comment

All pay auctions with certain and uncertain prizes a comment CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No. 1-2015 All py uctions with certin nd uncertin prizes comment Christin Riis All py uctions with certin nd uncertin prizes comment Christin

More information

4.11 Inner Product Spaces

4.11 Inner Product Spaces 314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces

More information

Factoring Polynomials

Factoring Polynomials Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles

More information

Net Change and Displacement

Net Change and Displacement mth 11, pplictions motion: velocity nd net chnge 1 Net Chnge nd Displcement We hve seen tht the definite integrl f (x) dx mesures the net re under the curve y f (x) on the intervl [, b] Any prt of the

More information

On the Meaning of Regression Coefficients for Categorical and Continuous Variables: Model I and Model II; Effect Coding and Dummy Coding

On the Meaning of Regression Coefficients for Categorical and Continuous Variables: Model I and Model II; Effect Coding and Dummy Coding Dt_nlysisclm On the Mening of Regression for tegoricl nd ontinuous Vribles: I nd II; Effect oding nd Dummy oding R Grdner Deprtment of Psychology This describes the simple cse where there is one ctegoricl

More information

Basic Analysis of Autarky and Free Trade Models

Basic Analysis of Autarky and Free Trade Models Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently

More information

Exponentiation: Theorems, Proofs, Problems Pre/Calculus 11, Veritas Prep.

Exponentiation: Theorems, Proofs, Problems Pre/Calculus 11, Veritas Prep. Exponentition: Theorems, Proofs, Problems Pre/Clculus, Verits Prep. Our Exponentition Theorems Theorem A: n+m = n m Theorem B: ( n ) m = nm Theorem C: (b) n = n b n ( ) n n Theorem D: = b b n Theorem E:

More information

Module Summary Sheets. C3, Methods for Advanced Mathematics (Version B reference to new book) Topic 2: Natural Logarithms and Exponentials

Module Summary Sheets. C3, Methods for Advanced Mathematics (Version B reference to new book) Topic 2: Natural Logarithms and Exponentials MEI Mthemtics in Ection nd Instry Topic : Proof MEI Structured Mthemtics Mole Summry Sheets C, Methods for Anced Mthemtics (Version B reference to new book) Topic : Nturl Logrithms nd Eponentils Topic

More information

Chapter 8 - Practice Problems 1

Chapter 8 - Practice Problems 1 Chpter 8 - Prctice Problems 1 MULTIPLE CHOICE. Choose the one lterntive tht best completes the sttement or nswers the question. A hypothesis test is to be performed. Determine the null nd lterntive hypotheses.

More information

Quadratic Equations - 1

Quadratic Equations - 1 Alger Module A60 Qudrtic Equtions - 1 Copyright This puliction The Northern Alert Institute of Technology 00. All Rights Reserved. LAST REVISED Novemer, 008 Qudrtic Equtions - 1 Sttement of Prerequisite

More information

An Undergraduate Curriculum Evaluation with the Analytic Hierarchy Process

An Undergraduate Curriculum Evaluation with the Analytic Hierarchy Process An Undergrdute Curriculum Evlution with the Anlytic Hierrchy Process Les Frir Jessic O. Mtson Jck E. Mtson Deprtment of Industril Engineering P.O. Box 870288 University of Albm Tuscloos, AL. 35487 Abstrct

More information

Curve Sketching. 96 Chapter 5 Curve Sketching

Curve Sketching. 96 Chapter 5 Curve Sketching 96 Chpter 5 Curve Sketching 5 Curve Sketching A B A B A Figure 51 Some locl mximum points (A) nd minimum points (B) If (x, f(x)) is point where f(x) reches locl mximum or minimum, nd if the derivtive of

More information

SPECIAL PRODUCTS AND FACTORIZATION

SPECIAL PRODUCTS AND FACTORIZATION MODULE - Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come

More information

Newton s Three Laws. d dt F = If the mass is constant, this relationship becomes the familiar form of Newton s Second Law: dv dt

Newton s Three Laws. d dt F = If the mass is constant, this relationship becomes the familiar form of Newton s Second Law: dv dt Newton s Three Lws For couple centuries before Einstein, Newton s Lws were the bsic principles of Physics. These lws re still vlid nd they re the bsis for much engineering nlysis tody. Forml sttements

More information

Plotting and Graphing

Plotting and Graphing Plotting nd Grphing Much of the dt nd informtion used by engineers is presented in the form of grphs. The vlues to be plotted cn come from theoreticl or empiricl (observed) reltionships, or from mesured

More information

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers. 2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this

More information

4.0 5-Minute Review: Rational Functions

4.0 5-Minute Review: Rational Functions mth 130 dy 4: working with limits 1 40 5-Minute Review: Rtionl Functions DEFINITION A rtionl function 1 is function of the form y = r(x) = p(x) q(x), 1 Here the term rtionl mens rtio s in the rtio of two

More information

DlNBVRGH + Sickness Absence Monitoring Report. Executive of the Council. Purpose of report

DlNBVRGH + Sickness Absence Monitoring Report. Executive of the Council. Purpose of report DlNBVRGH + + THE CITY OF EDINBURGH COUNCIL Sickness Absence Monitoring Report Executive of the Council 8fh My 4 I.I...3 Purpose of report This report quntifies the mount of working time lost s result of

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology

More information

Bayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Bayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology

More information

Helicopter Theme and Variations

Helicopter Theme and Variations Helicopter Theme nd Vritions Or, Some Experimentl Designs Employing Pper Helicopters Some possible explntory vribles re: Who drops the helicopter The length of the rotor bldes The height from which the

More information

Graphs on Logarithmic and Semilogarithmic Paper

Graphs on Logarithmic and Semilogarithmic Paper 0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology

More information

Section 2.3. Motion Along a Curve. The Calculus of Functions of Several Variables

Section 2.3. Motion Along a Curve. The Calculus of Functions of Several Variables The Clculus of Functions of Severl Vribles Section 2.3 Motion Along Curve Velocity ccelertion Consider prticle moving in spce so tht its position t time t is given by x(t. We think of x(t s moving long

More information

Chapter 9: Quadratic Equations

Chapter 9: Quadratic Equations Chpter 9: Qudrtic Equtions QUADRATIC EQUATIONS DEFINITION + + c = 0,, c re constnts (generlly integers) ROOTS Synonyms: Solutions or Zeros Cn hve 0, 1, or rel roots Consider the grph of qudrtic equtions.

More information

11. Fourier series. sin mx cos nx dx = 0 for any m, n, sin 2 mx dx = π.

11. Fourier series. sin mx cos nx dx = 0 for any m, n, sin 2 mx dx = π. . Fourier series Summry of the bsic ides The following is quick summry of the introductory tretment of Fourier series in MATH. We consider function f with period π, tht is, stisfying f(x + π) = f(x) for

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology

More information

EQUATIONS OF LINES AND PLANES

EQUATIONS OF LINES AND PLANES EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in point-direction nd twopoint

More information

The Velocity Factor of an Insulated Two-Wire Transmission Line

The Velocity Factor of an Insulated Two-Wire Transmission Line The Velocity Fctor of n Insulted Two-Wire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the

More information

Mathematics Higher Level

Mathematics Higher Level Mthemtics Higher Level Higher Mthemtics Exmintion Section : The Exmintion Mthemtics Higher Level. Structure of the exmintion pper The Higher Mthemtics Exmintion is divided into two ppers s detiled below:

More information

Lecture 3 Basic Probability and Statistics

Lecture 3 Basic Probability and Statistics Lecture 3 Bsic Probbility nd Sttistics The im of this lecture is to provide n extremely speedy introduction to the probbility nd sttistics which will be needed for the rest of this lecture course. The

More information

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one. 5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous rel-vlued

More information

Anti-derivatives/Indefinite Integrals of Basic Functions

Anti-derivatives/Indefinite Integrals of Basic Functions Anti-derivtives/Indefinite Integrls of Bsic Functions Power Rule: x n+ x n n + + C, dx = ln x + C, if n if n = In prticulr, this mens tht dx = ln x + C x nd x 0 dx = dx = dx = x + C Integrl of Constnt:

More information

Answer, Key Homework 8 David McIntyre 1

Answer, Key Homework 8 David McIntyre 1 Answer, Key Homework 8 Dvid McIntyre 1 This print-out should hve 17 questions, check tht it is complete. Multiple-choice questions my continue on the net column or pge: find ll choices before mking your

More information

Section 5-4 Trigonometric Functions

Section 5-4 Trigonometric Functions 5- Trigonometric Functions Section 5- Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form

More information

MATLAB: M-files; Numerical Integration Last revised : March, 2003

MATLAB: M-files; Numerical Integration Last revised : March, 2003 MATLAB: M-files; Numericl Integrtion Lst revised : Mrch, 00 Introduction to M-files In this tutoril we lern the bsics of working with M-files in MATLAB, so clled becuse they must use.m for their filenme

More information

Review guide for the final exam in Math 233

Review guide for the final exam in Math 233 Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered

More information

Economics Letters 65 (1999) 9 15. macroeconomists. a b, Ruth A. Judson, Ann L. Owen. Received 11 December 1998; accepted 12 May 1999

Economics Letters 65 (1999) 9 15. macroeconomists. a b, Ruth A. Judson, Ann L. Owen. Received 11 December 1998; accepted 12 May 1999 Economics Letters 65 (1999) 9 15 Estimting dynmic pnel dt models: guide for q mcroeconomists b, * Ruth A. Judson, Ann L. Owen Federl Reserve Bord of Governors, 0th & C Sts., N.W. Wshington, D.C. 0551,

More information

Generalized Inverses: How to Invert a Non-Invertible Matrix

Generalized Inverses: How to Invert a Non-Invertible Matrix Generlized Inverses: How to Invert Non-Invertible Mtrix S. Swyer September 7, 2006 rev August 6, 2008. Introduction nd Definition. Let A be generl m n mtrix. Then nturl question is when we cn solve Ax

More information

Small Business Networking

Small Business Networking Why Network is n Essentil Productivity Tool for Any Smll Business TechAdvisory.org SME Reports sponsored by Effective technology is essentil for smll businesses looking to increse their productivity. Computer

More information

A.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324

A.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324 A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................

More information

Reasoning to Solve Equations and Inequalities

Reasoning to Solve Equations and Inequalities Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing

More information

2005-06 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration

2005-06 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration Source: http://www.mth.cuhk.edu.hk/~mt26/mt26b/notes/notes3.pdf 25-6 Second Term MAT26B 1 Supplementry Notes 3 Interchnge of Differentition nd Integrtion The theme of this course is bout vrious limiting

More information

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn 33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of

More information

Math 135 Circles and Completing the Square Examples

Math 135 Circles and Completing the Square Examples Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for

More information

Algebra Review. How well do you remember your algebra?

Algebra Review. How well do you remember your algebra? Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then

More information

Math Review 1. , where α (alpha) is a constant between 0 and 1, is one specific functional form for the general production function.

Math Review 1. , where α (alpha) is a constant between 0 and 1, is one specific functional form for the general production function. Mth Review Vribles, Constnts nd Functions A vrible is mthemticl bbrevition for concept For emple in economics, the vrible Y usully represents the level of output of firm or the GDP of n economy, while

More information

r 2 F ds W = r 1 qe ds = q

r 2 F ds W = r 1 qe ds = q Chpter 4 The Electric Potentil 4.1 The Importnt Stuff 4.1.1 Electricl Potentil Energy A chrge q moving in constnt electric field E experiences force F = qe from tht field. Also, s we know from our study

More information

AP Statistics Testbank 7

AP Statistics Testbank 7 AP Sttistics Testbnk 7 Multiple-Choice Questions 1) In formulting hypotheses for sttisticl test of significnce, the null hypothesis is often ) sttement of "no effect" or "no difference." b) the probbility

More information

Integration by Substitution

Integration by Substitution Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is

More information

Algorithms Chapter 4 Recurrences

Algorithms Chapter 4 Recurrences Algorithms Chpter 4 Recurrences Outline The substitution method The recursion tree method The mster method Instructor: Ching Chi Lin 林清池助理教授 chingchilin@gmilcom Deprtment of Computer Science nd Engineering

More information

Section 7-4 Translation of Axes

Section 7-4 Translation of Axes 62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 7-4 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the

More information

Theory of Forces. Forces and Motion

Theory of Forces. Forces and Motion his eek extbook -- Red Chpter 4, 5 Competent roblem Solver - Chpter 4 re-lb Computer Quiz ht s on the next Quiz? Check out smple quiz on web by hurs. ht you missed on first quiz Kinemtics - Everything

More information

Regular Sets and Expressions

Regular Sets and Expressions Regulr Sets nd Expressions Finite utomt re importnt in science, mthemtics, nd engineering. Engineers like them ecuse they re super models for circuits (And, since the dvent of VLSI systems sometimes finite

More information

1. The leves re either lbeled with sentences in ;, or with sentences of the form All X re X. 2. The interior leves hve two children drwn bove them) if

1. The leves re either lbeled with sentences in ;, or with sentences of the form All X re X. 2. The interior leves hve two children drwn bove them) if Q520 Notes on Nturl Logic Lrry Moss We hve seen exmples of wht re trditionlly clled syllogisms lredy: All men re mortl. Socrtes is mn. Socrtes is mortl. The ide gin is tht the sentences bove the line should

More information

A new algorithm for generating Pythagorean triples

A new algorithm for generating Pythagorean triples A new lgorithm for generting Pythgoren triples RH Dye 1 nd RWD Nicklls 2 The Mthemticl Gzette (1998); 82 (Mrch, No. 493), p. 86 91 (JSTOR rchive) http://www.nicklls.org/dick/ppers/mths/pythgtriples1998.pdf

More information

Binary Representation of Numbers Autar Kaw

Binary Representation of Numbers Autar Kaw Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse- rel number to its binry representtion,. convert binry number to n equivlent bse- number. In everydy

More information

FUNCTIONS AND EQUATIONS. xεs. The simplest way to represent a set is by listing its members. We use the notation

FUNCTIONS AND EQUATIONS. xεs. The simplest way to represent a set is by listing its members. We use the notation FUNCTIONS AND EQUATIONS. SETS AND SUBSETS.. Definition of set. A set is ny collection of objects which re clled its elements. If x is n element of the set S, we sy tht x belongs to S nd write If y does

More information

CHAPTER 11 Numerical Differentiation and Integration

CHAPTER 11 Numerical Differentiation and Integration CHAPTER 11 Numericl Differentition nd Integrtion Differentition nd integrtion re bsic mthemticl opertions with wide rnge of pplictions in mny res of science. It is therefore importnt to hve good methods

More information

N Mean SD Mean SD Shelf # Shelf # Shelf #

N Mean SD Mean SD Shelf # Shelf # Shelf # NOV xercises smple of 0 different types of cerels ws tken from ech of three grocery store shelves (1,, nd, counting from the floor). summry of the sugr content (grms per serving) nd dietry fiber (grms

More information

PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1

PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1 PROBLEMS - APPLICATIONS OF DERIVATIVES Pge ( ) Wter seeps out of conicl filter t the constnt rte of 5 cc / sec. When the height of wter level in the cone is 5 cm, find the rte t which the height decreses.

More information

JaERM Software-as-a-Solution Package

JaERM Software-as-a-Solution Package JERM Softwre-s--Solution Pckge Enterprise Risk Mngement ( ERM ) Public listed compnies nd orgnistions providing finncil services re required by Monetry Authority of Singpore ( MAS ) nd/or Singpore Stock

More information

AREA OF A SURFACE OF REVOLUTION

AREA OF A SURFACE OF REVOLUTION AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.

More information

Biostatistics 102: Quantitative Data Parametric & Non-parametric Tests

Biostatistics 102: Quantitative Data Parametric & Non-parametric Tests Singpore Med J 2003 Vol 44(8) : 391-396 B s i c S t t i s t i c s F o r D o c t o r s Biosttistics 102: Quntittive Dt Prmetric & Non-prmetric Tests Y H Chn In this rticle, we re going to discuss on the

More information

An Off-Center Coaxial Cable

An Off-Center Coaxial Cable 1 Problem An Off-Center Coxil Cble Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Nov. 21, 1999 A coxil trnsmission line hs inner conductor of rdius nd outer conductor

More information

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mthemtics Bsic Alger. Opertions nd Epressions. Common Mistkes. Division of Algeric Epressions. Eponentil Functions nd Logrithms. Opertions nd their Inverses. Mnipulting

More information

Continuous probability distributions

Continuous probability distributions Chpter 8 Continuous probbility distributions 8.1 Introduction In Chpter 7, we explored the concepts of probbility in discrete setting, where outcomes of n experiment cn tke on only one of finite set of

More information

MATLAB Workshop 13 - Linear Systems of Equations

MATLAB Workshop 13 - Linear Systems of Equations MATLAB: Workshop - Liner Systems of Equtions pge MATLAB Workshop - Liner Systems of Equtions Objectives: Crete script to solve commonly occurring problem in engineering: liner systems of equtions. MATLAB

More information

Basic Math Review. Numbers. Important Properties. Absolute Value PROPERTIES OF ADDITION NATURAL NUMBERS {1, 2, 3, 4, 5, }

Basic Math Review. Numbers. Important Properties. Absolute Value PROPERTIES OF ADDITION NATURAL NUMBERS {1, 2, 3, 4, 5, } ƒ Bsic Mth Review Numers NATURAL NUMBERS {1,, 3, 4, 5, } WHOLE NUMBERS {0, 1,, 3, 4, } INTEGERS {, 3,, 1, 0, 1,, } The Numer Line 5 4 3 1 0 1 3 4 5 Negtive integers Positive integers RATIONAL NUMBERS All

More information

CS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001

CS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001 CS99S Lortory 2 Preprtion Copyright W. J. Dlly 2 Octoer, 2 Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes to oserve logic

More information

According to Webster s, the

According to Webster s, the dt modeling Universl Dt Models nd P tterns By Len Silversn According Webster s, term universl cn be defined s generlly pplicble s well s pplying whole. There re some very common ptterns tht cn be generlly

More information

Corporate Compliance vs. Enterprise-Wide Risk Management

Corporate Compliance vs. Enterprise-Wide Risk Management Corporte Complince vs. Enterprise-Wide Risk Mngement Brent Sunders, Prtner (973) 236-4682 November 2002 Agend Corporte Complince Progrms? Wht is Enterprise-Wide Risk Mngement? Key Differences Why Will

More information