ETSI TS V1.2.1 ( )

Transcription

1 TS V.. (004-0) Techcal Specfcato Speech Processg, Trasmsso ad Qualty Aspects (STQ); QoS aspects for popular servces GSM ad 3G etworks; Part 6: Post processg ad statstcal methods

2 TS V.. (004-0) Referece RTS/STQ-0006m Keywords 3G, GSM, etwork, QoS, servce, speech 650 Route des Lucoles F-069 Sopha Atpols Cede - FRANCE Tel.: Fa: Sret N NAF 74 C Assocato à but o lucratf eregstrée à la Sous-Préfecture de Grasse (06) N 7803/88 Importat otce Idvdual copes of the preset documet ca be dowloaded from: The preset documet may be made avalable more tha oe electroc verso or prt. I ay case of estg or perceved dfferece cotets betwee such versos, the referece verso s the Portable Documet Format (PDF). I case of dspute, the referece shall be the prtg o prters of the PDF verso kept o a specfc etwork drve wth Secretarat. Users of the preset documet should be aware that the documet may be subject to revso or chage of status. Iformato o the curret status of ths ad other documets s avalable at If you fd errors the preset documet, please sed your commet to oe of the followg servces: Copyrght Notfcato No part may be reproduced ecept as authorzed by wrtte permsso. The copyrght ad the foregog restrcto eted to reproducto all meda. Europea Telecommucatos Stadards Isttute 004. All rghts reserved. DECT TM, PLUGTESTS TM ad UMTS TM are Trade Marks of regstered for the beeft of ts Members. TIPHON TM ad the TIPHON logo are Trade Marks curretly beg regstered by for the beeft of ts Members. 3GPP TM s a Trade Mark of regstered for the beeft of ts Members ad of the 3GPP Orgazatoal Parters.

3 3 TS V.. (004-0) Cotets Itellectual Property Rghts...6 Foreword...6 Itroducto...7 Scope...8 Refereces Deftos, symbols ad abbrevatos Deftos Symbols Abbrevatos Importat measuremet data types moble commucatos Data wth bary values Data out of tme-terval measuremets Measuremet of data throughput Data cocerg qualty measures Dstrbutos ad momets Itroducto Cotuous ad dscrete dstrbutos Defto of desty fucto ad dstrbuto fucto Probablty Dstrbuto Fucto (PDF) Cumulatve Dstrbuto Fucto (CDF) Momets ad quatles Estmato of momets ad quatles Importat dstrbutos Cotuous dstrbutos Normal dstrbuto Stadard ormal dstrbuto Cetral lmt theorem Trasformato to ormalty Log-Normal dstrbuto Use-case: trasformatos Epoetal dstrbuto Webull dstrbuto Pareto dstrbuto Etreme dstrbuto (Fsher-Tppett dstrbuto) Testg dstrbutos Ch-Square dstrbuto wth degrees of freedom Further relatos Relato to emprcal varace Studet t-dstrbuto Relato to ormal dstrbuto F dstrbuto Quatles Appromato of quatles Relatos to other dstrbutos Dscrete dstrbutos Beroull dstrbuto Bomal dstrbuto Geometrc dstrbuto Posso dstrbuto Trastos betwee dstrbutos ad approprate appromatos From bomal to Posso dstrbuto From bomal to Normal dstrbuto From Posso to Normal dstrbuto...3

4 4 TS V.. (004-0) Trucated dstrbutos Dstrbuto selecto ad parameter estmato Test procedures Ch-Square test Kolmogorov-Smrov test Shapro-Wlk test Parameter estmato methods Evaluato of measuremet data Statstcal tests Formulato of statstcal tests Classes of statstcal tests Tests for ormal ad bomal data Oe-sample tests for ormal data Two-sample tests for ormal data Test for bomal data Dstrbuto-free tests for locato Sg tests Sg rak test Wlcoo rak sum test Cofdece terval Bomal dstrbuto Normal (Gaussa) dstrbuto Requred sample sze for certa cofdece levels Vsualzato techques Vsualzato of statc data Hstograms Barplots QQ-Plots Boplots Vsualzato of dyamc data Le Dagrams Temporal chagg Boplots MMQ-Plots Tme seres modellg Descrptve characterzato Emprcal momets Decomposto of tme seres Determato of the tred compoet Tred fucto types Lear tred fucto Polyomal tred fucto No-lear tred models Tred estmato Trasformato of tme seres by flterg Lear flters Epoetal flters Seasoal compoet Data aggregato Basc data aggregato operators Data sources, structures ad propertes Raw data Performace data Evet data Key Performace Idcators / Parameters Aggregato herarches Temporal aggregato Spatal aggregato Parameter estmato methods Projecto method Substtuto method...6

5 5 TS V.. (004-0) Applcato of estmato methods Attrbutes of aggregato operators Weghted aggregato Perceved QoS Weghted quatles Addtoal data aggregato operators MAWD ad BH AVG Assessmet of performace dces Estmato of performace parameters based o actve servce probg systems Motorg cocepts Cotrol charts Shewhart cotrol charts CUSUM ad EWMA charts Other alarmg rules Methods for evaluato of objectves Desrablty fuctos Loss fuctos...67 Ae A (formatve): Eamples of statstcal calculatos...68 A. Cofdece tervals for bomal dstrbuto...68 A.. Step by step computato...68 A.. Computato usg statstcal software...70 A... Computato R...70 A... Computato Ecel...7 A. Trasto from bomal to ormal dstrbuto...7 A.3 Deftos of EG A.4 Calculato of cofdece tervals...73 A.4. Estmated rate 5 %...73 A.4. Estmated rate 50 %...74 A.4.3 Estmated rate 95 %...74 A.4.4 Lower lmt of cofdece tervals accordg to Pearso-Clopper formula...75 A.4.5 Upper lmt of cofdece tervals accordg to Pearso-Clopper formula...76 A.4.6 Spa of cofdece tervals accordg to Pearso-Clopper formula...78 A.5 Dfferet sample szes...80 A.6 Calculato methods...8 A.6. Calculato of quatles...8 A.7 Reportg of results...84 A.7. Methods to use...84 A.7. Number of sgfcat decmals...85 A.7.3 Roudg of ed results...85 Ae B (formatve): Bblography...86 Hstory...87

6 6 TS V.. (004-0) Itellectual Property Rghts IPRs essetal or potetally essetal to the preset documet may have bee declared to. The formato pertag to these essetal IPRs, f ay, s publcly avalable for members ad o-members, ad ca be foud SR : "Itellectual Property Rghts (IPRs); Essetal, or potetally Essetal, IPRs otfed to respect of stadards", whch s avalable from the Secretarat. Latest updates are avalable o the Web server ( Pursuat to the IPR Polcy, o vestgato, cludg IPR searches, has bee carred out by. No guaratee ca be gve as to the estece of other IPRs ot refereced SR (or the updates o the Web server) whch are, or may be, or may become, essetal to the preset documet. Foreword Ths Techcal Specfcato (TS) has bee produced by Techcal Commttee Speech Processg, Trasmsso ad Qualty Aspects (STQ). The preset documet s part 6 of a mult-part delverable coverg the QoS aspects for popular servces GSM ad 3G etworks, as detfed below: Part : Part : Part 3: Part 4: Part 5: Part 6: Part 7: "Idetfcato of Qualty of Servce aspects"; "Defto of Qualty of Servce parameters ad ther computato"; "Typcal procedures for Qualty of Servce measuremet equpmet"; "Requremets for Qualty of Servce measuremet equpmet"; "Defto of typcal measuremet profles"; "Post processg ad statstcal methods"; "Samplg methodology". Part detfes QoS aspects for popular servces GSM ad 3G etworks. For each servce chose QoS dcators are lsted. They are cosdered to be sutable for the quattatvely characterzato of the domat techcal QoS aspects as epereced from the ed-customer perspectve. Part defes QoS parameters ad ther computato for popular servces GSM ad 3G etworks. The techcal QoS dcators, lsted part, are the bass for the parameter set chose. The parameter defto s splt to two parts: the abstract defto ad the geerc descrpto of the measuremet method wth the respectve trgger pots. Oly measuremet methods ot depedet o ay frastructure provded are descrbed the preset documet. The harmozed deftos gve the preset documet are cosdered as the prerequstes for comparso of QoS measuremets ad measuremet results. Part 3 descrbes typcal procedures used for QoS measuremets over GSM, alog wth settgs ad parameters for such measuremets. Part 4 defes the mmum requremets of QoS measuremet equpmet for GSM ad 3G etworks the way that the values ad trgger-pots eeded to compute the QoS parameter as defed part ca be measured followg the procedures defed part 3. Test-equpmet fulfllg the specfed mmum requremets, wll allow to perform the proposed measuremets a relable ad reproducble way. Part 5 specfes test profles whch are requred to eable bechmarkg of dfferet GSM or 3G etworks both wth ad outsde atoal boudares. It s ecessary to have these profles so that whe a specfc set of tests are carred out the customers are comparg "lke for lke" performace.

7 7 TS V.. (004-0) Part 6 descrbes procedures to be used for statstcal calculatos the feld of QoS measuremet of GSM ad 3G etwork usg probg systems. Part 7 descrbes the feld measuremet method procedures used for QoS measuremets over GSM where the results are obtaed applyg feretal statstcs. Itroducto All the defed qualty of servce parameters ad ther computatos are based o feld measuremets. That dcates that the measuremets were made from customers pot of vew (full ed-to-ed perspectve, takg to accout the eeds of testg). It s assumed that the ed customer ca hadle hs moble ad the servces he wats to use (operablty s ot evaluated at ths tme). For the purpose of measuremet t s assumed: that the servce s avalable ad ot barred for ay reaso; routg s defed correctly wthout errors; ad the target subscrber equpmet s ready to aswer the call. Voce qualty values measured should oly be employed by calls eded successfully for statstcal aalyss. However, measured values from calls eded usuccessfully (e.g. dropped) should be avalable for addtoal evaluatos ad therefore, must be stored. Further precodtos may apply whe reasoable.

8 8 TS V.. (004-0) Scope The preset documet descrbes deftos ad procedures to be used for statstcal calculatos whch are related to Qualty of Servce (QoS) measuremets doe by servg probg systems moble commucatos etworks, especally GSM ad 3G etworks. Network performace measuremets ad ther related post-processg are oly margally covered the preset documet. Refereces The followg documets cota provsos whch, through referece ths tet, costtute provsos of the preset documet. Refereces are ether specfc (detfed by date of publcato ad/or edto umber or verso umber) or o-specfc. For a specfc referece, subsequet revsos do ot apply. For a o-specfc referece, the latest verso apples. Refereced documets whch are ot foud to be publcly avalable the epected locato mght be foud at [] EG 0 769: "Speech Processg, Trasmsso ad Qualty Aspects (STQ); QoS parameter deftos ad measuremets; Parameters for voce telephoy servce requred uder the ONP Voce Telephoy Drectve 98/0/EC". 3 Deftos, symbols ad abbrevatos 3. Deftos For the purposes of the preset documet, the followg terms ad deftos apply: rate: measuremet result whch s related to the porto of tme durg whch t has bee eecuted NOTE: The deomator's ut s related to tme. rato: measuremet result whch represets a subgroup of all sgle measuremets s related to the total umber of eecuted sgle measuremets NOTE: Usually, omator ad deomator share the same ut, amely a couter for measuremets (subgroup/all).

9 9 TS V.. (004-0) 3. Symbols For the purposes of the preset documet, the followg symbols apply: E()µ Epected value of radom varable Var()σ Varace of radom varable σ Stadard devato of radom varable f() Probablty Desty Fucto (PDF) of radom varable F() Cumulatve Dstrbuto Fucto (CDF) of radom varable S, S Set of dscrete values or terval of values the radom varable may take IR Set of real umbers s, s Emprcal stadard devato / varace, aalogous to σ ad σ (theoretcal) q α α-quatle u α α-quatle of stadard ormal dstrbuto (), (), () -th ordered value, mmum ad mamum of a gve data set,,..., 3.3 Abbrevatos For the purposes of the preset documet, the followg abbrevatos apply: 3G ARMA AVG BH BSC CDF CUSUM EWMA GSM KPI LSL MAWD MMQ-Plot MMS MOS MSC NE PDF QoS QQ-Plot SMS USL Thrd Geerato Auto-Regressve Movg Average Averagg Operator (regardg days) Busy Hour Base Stato Cotroller Cumulatve Dstrbuto Fucto or Cumulatve Desty Fucto (used syoymously) CUmulated SUM Epoetally Weghted Movg Average Global System for Moble commucatos Key Performace Idcator Lower Specfcato Level Mothly Average Workg Day Meda-Mea-Quatle Plot Multmeda Messagg Servce Mea Opo Score Moble Swtchg Cetre Network Elemet Probablty Desty Fucto Qualty of Servce Quatle-Quatle Plot Short Message Servce Upper Specfcato Level 4 Importat measuremet data types moble commucatos Approprate data aalyss methods should deped o the type of the gve data as well as o the scope of the aalyss. Therefore before aalyss methods are descrbed, dfferet data types are troduced ad dffereces betwee them are poted out. Four geeral categores of measuremet results are epected whe QoS measuremets are doe moble commucatos.

10 0 TS V.. (004-0) 4. Data wth bary values Sgle measuremets related to the topcs: servce accessblty, servce avalablty; servce retaablty, servce cotuty; error ratos, error probabltes; geeral show a bary outcome,.e. oly two outcomes are possble. Ths meas the result of a sgle tral leads to a result whch s ether valued postve or egatve related to the cosdered objectve. The result may be recorded as decso-results Yes / No or True / False or wth umercal values 0 successful ad usuccessful (.e. errors occur) or vce versa. Aggregato of trals of both types allows to calculate the accordg ratos whch meas the umber of postve / egatve results s dvded by the umber of all trals. Usually, the uts of omator ad deomator are the same, amely umber of trals. EXAMPLE: If establshed voce calls are cosdered to test the servce retaablty of a voce telephoy system, every successfully completed call leads to the postve result "Call completed", every usuccessfully eded call s otced as "Dropped call" whch represets the egatve outcome. After establshed calls, the rato of dropped calls related to all establshed calls ca be calculated. The result s the call drop probablty. 4. Data out of tme-terval measuremets Measuremets related to the tme doma occur the areas: durato of a sesso or call; servce access delay; roud trp tme ad ed-to-ed delay of a servce; blockg tmes, dowtmes of a system. The outcome of such measuremets s the tme spa betwee two tme stamps markg the startg ad ed pot of the tme perods of terest. Results are related to the ut "secod" or multples or parts of t. Depedg o the measuremet tools ad the precso eeded, arbtrarly small measuremet uts may be realzed. EXAMPLE: Someoe ca defe the ed-to-ed delvery tme for the MMS servce by a measuremet whch starts whe the user at the A party pushes the "Sed" butto ad whch stops whe the completely receved MMS s sgalled to the user at the B party. 4.3 Measuremet of data throughput Measuremets related to data throughput result values whch descrbe the rato of trasmtted data volume related to the requred porto of tme. The outcome of a sgle measuremet s the quotet of both measures. Used uts are "bt" or multples thereof for the data amout ad "secod" or multples or parts thereof for the porto of tme. EXAMPLE: If a data amout of Mbt s trasmtted wth a perod of 60 secods, ths results a mea data rate of appromately 6,66 kbt/s. 4.4 Data cocerg qualty measures Eamples are gve by the qualty of data trasfer whch may be measured by ts speed or evaluatos of speech qualty measured o a scale, respectvely.

11 TS V.. (004-0) Measuremets related to audo-vsual qualty ca be doe objectvely by algorthms or subjectvely by huma lsteers. The outcome of audo-vsual qualty evaluato s related to a scaled value whch s called Mea Opo Score (MOS) for subjectve testg. Thereby two types of qualty measuremet are dstgushed subjectve ad objectve measuremets. If quattatve measures are detfed whch are hghly correlated to the qualty of terest, ths wll smplfy the aalyss. However, f ths s ot possble, some kd of evaluato o a stadardzed scale by qualfed eperts s eeded. The result may therefore be gve ether as the measuremet result or as a mark o a pre-defed scale. EXAMPLE: Wth a subjectve test, people are asked to rate the overall qualty of vdeo samples whch are preseted to them. The allowed scale to rate the qualty s defed the rage from (very poor qualty) to 5 (brllat qualty). Table 4. summarzes the dfferet kds of QoS related measuremets, typcal outcomes ad some eamples. Table 4.: QoS related measuremets, typcal outcomes ad eamples Category Relevat measuremet types Eamples Bary values Servce accessblty, servce avalablty Servce retaablty, servce cotuty Error ratos, error probabltes Servce accessblty telephoy, servce o-avalablty SMS Call completo rate, call drop rate Call set-up error rate Durato values Durato of a sesso or call Servce access delay Roud trp tme, ed-to-ed delay Blockg tmes, system dowtmes Mea call durato Servce access delay WAP ICMP Pg roudtrp tme Blockg tme telephoy, SGSN dowtme Throughput values Throughput Mea data rate GPRS Peak data rate UMTS Cotet qualty values Audo-vsual qualty MOS scores out of subjectve testg 5 Dstrbutos ad momets 5. Itroducto The objectve of data aalyses s to draw coclusos about the state of a process based o a gve data set, whch may or may ot be a sample of the populato of terest. If dstrbutos are assumed, these specfy the shape of the data mass up to parameters assocated wth each famly of dstrbutos specfyg propertes lke the mea of the data mass. Locato or dsperso shfts of the process wll geeral result dfferet parameter estmates specfyg the dstrbuto. Therefore the formato avalable from the data s compressed to oe or few suffcet statstcs specfyg the uderlyg dstrbuto. May statstcal applcatos ad computatos rely some sese o dstrbutoal assumptos, whch are ot always eplctly stated. Results of statstcal measures are ofte oly sesble f uderlyg assumptos are met ad therefore oly terpretable f users kow about these assumptos. Ths clause s orgazed as follows. Frstly, dstrbutos, momets ad quatles are troduced theory clauses 5. to 5.4. Ths part of the documet s based o the dea of radom varables havg certa dstrbutos. Radom varables do ot take sgle values but descrbe the uderlyg probablty model of a radom process. They are commoly deoted by: X ~ dstrbuto (parameters) From the dstrbutoal assumptos, momets ad quatles of radom varables are derved theory. Data s ofte vewed as beg realzatos of radom varables. Therefore, data aalyss maly cossts of fttg a approprate dstrbuto to the data ad drawg coclusos based o ths assumpto. Clause 5.5 brefly summarzes the estmato of momets ad quatles. Subsequetly, a umber of mportat dstrbutos s troduced clause 5.6, each of whch s vsualzed graphcally to gve a dea of meagful applcatos. Wth ths clause, testg dstrbutos are also troduced as they are eeded clause 5.7 for the dervato of statstcal tests.

12 TS V.. (004-0) 5. Cotuous ad dscrete dstrbutos The ma dfferece betwee the data types descrbed above ca be eplaed terms of cotuous ad dscrete dstrbutos. Data wth bary values follow a dscrete dstrbuto, sce the probablty mass s dstrbuted oly over a fed umber of possble values. The same holds for qualty measuremets wth evaluato results o a scale wth a lmted umber of possble values (.e. marks to 6 or smlar). O the cotrary, tme-terval measuremets as well as qualty measuremets based o approprate quattatve varables may take a ftely large umber of possble values. I theory, sce the umber of possble outcomes equals fty, the probablty that a sgle value s eactly realzed s zero. Probabltes greater tha zero are oly realzed for tervals wth postve wdth. I practce, each measuremet tool wll oly allow a lmted precso resultg dscrete measuremets wth a large umber of possble outcomes. Nevertheless, data from measuremet systems wth reasoable precso are treated as beg cotuous. Formal deftos for cotuous ad dscrete dstrbutos are based o probablty desty fuctos as wll be descrbed the followg. 5.3 Defto of desty fucto ad dstrbuto fucto 5.3. Probablty Dstrbuto Fucto (PDF) Probablty Desty Fuctos (PDF) specfy the probablty mass ether for sgle outcomes (dscrete dstrbutos) or for tervals (cotuous dstrbutos). A PDF s defed as a fucto : IR [ 0, ) ) f ( ) 0 for all S. f wth propertes: ) f ( ) d for cotuous dstrbutos or f ( ) for dscrete dstrbutos. S S I other words, frstly the values of the PDF are always o-egatve, meag that egatve probabltes are ether assged to values or tervals, ad secodly the summato or tegrato over the PDF always results ( 00 %), meag that ay data value wll always be realzed. EXAMPLE : EXAMPLE : EXAMPLE 3: 0,: A PDF for bary data may be gve by f (), whch mples that the probablty for 0,9 : 0 a faulty tral () s 0 %, whle tests are completed successfully wth probablty 90 %. For tme-terval measuremets PDFs may take ay kd of shape, as a eample a ormal dstrbuto wth mea 0 (secods) s assumed here. The PDF for ths dstrbuto s gve by π { ( ) } f ( ) ep 0. Other eamples for cotuous dstrbutos wll follow later o. If for stace categores for speech qualty are defed as very poor up to 5 brllat, a PDF 0, : {,,3} for the resultg data may be gve by f ( ) 0,4 : 4. 0,3 : 5 Fgure 5. summarzes all three assumed eample PDFs for the dfferet data types.

13 3 TS V.. (004-0) Eample Eample Eample 3 f() f() f() Fgure 5.: Probablty Desty Fuctos (PDFs) of eamples to Cumulatve Dstrbuto Fucto (CDF) A Cumulatve Dstrbuto (or Desty) Fucto (CDF) s computed from the correspodg PDF as descrbed before by summg (dscrete) or tegratg (cotuous) over the desty mass up to the curret value. F wth F ( ) ~ f ( ~ ) for dscrete ad F( ) f ( ~ ) d ~ for cotuous dstrbutos s called CDF. Ths mples F ( ) for ad F ( ) 0 for. A fucto : IR [ 0,] I other words, the value of the CDF correspods to the proporto of the dstrbuto left of the value of terest. For the three eamples from above, the CDFs are gve fgure 5.. Eample Eample Eample 3 F() F() F() Fgure 5.: Cumulatve Dstrbuto Fuctos (CDFs) of eamples to Momets ad quatles Momets are ma characterstcs of dstrbutos. The most mportat momets are: the epected value (frst momet), specfyg the locato of the dstrbuto; the varace (secod cetral momet), specfyg the dsperso aroud the epected value of the dstrbuto; ad the skewess (thrd cetral momet), specfyg whether a dstrbuto s symmetrc or skewed.

14 4 TS V.. (004-0) These momets are defed as follows. a) The epected value (frst momet, mea) of a radom varable wth CDF f() s defed as E ( ) f ( ) d for cotuous dstrbutos or E ( ) f ( ) for dscrete dstrbutos, respectvely. b) The varace (secod cetral momet) of a radom varable wth CDF f() s defed as ( E( ) ) Var ( ) f ( ) d for cotuous dstrbutos or ( ) ( E( ) ) Var f ( ) for dscrete dstrbutos, respectvely. The square root of the varace called stadard devato, deoted as σ() s ofte more formatve sce t s defed o the orgal data scale. c) The skewess (thrd cetral momet) of a radom varable wth CDF f() s defed as ( E( ) ) 3 f ( ) d E( ) f ( ) for dscrete dstrbutos, respectvely. A value of zero dcates a symmetrc dstrbuto. for cotuous dstrbutos or ( ) EXAMPLE : For the CDF from eample the momets are gve by E() 0, +0,9 0 0,, Var() 0, 0,9 +0,9 0, 0,09 resultg a stadard devato σ() 0,3. The skewess ca be computed as 0, 0, (-0,) 3 0,07 dcatg that the dstrbuto s ot symmetrc. 3 EXAMPLE : The momets of the above ormal dstrbuto ca be computed by partal tegrato ad the fact that the PDF tegrates to, or by utlzg the propertes of ormal dstrbutos statg that the mea ad stadard devato are the parameters µ ad σ of the PDF f ( µ, σ ) ep ( 0) ad that ormal dstrbutos are always symmetrc. σ π σ Ths results E() 0, Var() σ, whch also equals the stadard devato ad skewess 0 for the above eample. EXAMPLE 3: For the CDF of eample 3, momets are computed by E() 0, +0, +0, 3+0,4 4+0,3 5 3,7, Var(),6 ad egatve skewess of -,84. The momets are computable for all three eample PDFs. Nevertheless, they are ot always meagful. I partcular the thrd eample, the possble outcomes are "very poor" to "brllat", whch may be ordered ad amed to 5 as has bee doe before, but the epected value of 3,7 does ot have a strct meag. The same apples for hgher momets, sce the values of the varable of terest are ot quattatve, but ordered qualtatve. I case of o-symmetrc dstrbuted data, momets may ot be approprate for descrbg the dstrbuto of terest. A alteratve measure of locato s gve by the meda, whch ca be vewed as the pot cuttg the dstrbuto to halves, amely 50 % of the dstrbuto mass are smaller ad 50 % are larger tha the meda. More geerally, quatles are defed for each possble percetage. The α-quatle cuts the dstrbuto a part of α 00 % of the dstrbuto smaller tha ths value ad (-α) 00 % larger tha ths value. The meda as a specal case s also called 50 %-quatle. A formal defto of quatles s for stace gve by Mood, Graybll, Boes (974): "The α-quatle q α wth α (0,] s defed as the smallest umber q α satsfyg F (q α ) α (for α 0, the mmum value wth postve probablty or - s defed, respectvely)". Quatles are easest llustrated wth the eamples of CDFs gve above, compare fgure 5.3. For each CDF, the 5 %, 50 % ad 75 %-quatles are added to the correspodg plot.

15 5 TS V.. (004-0) Eample Eample Eample F() 0.50 F() 0.50 F() q 0.05 q 0.5, q 0.75 q 0.05 q 0.5 q 0.75 q 0.05 q 0.5 q 0.75 Fgure 5.3: Illustrato of theoretcal quatles for eamples to Estmato of momets ad quatles If oly samples from the populato of terest are avalable, theoretcal momets may ot be computed, but have to be estmated emprcally. A sample-based estmator of the epectato of the uderlyg dstrbuto s gve by the emprcal mea, where,,..., are the sample values. The varace of a dstrbuto s commoly estmated by s ( ) wth resultg emprcal stadard devato s ( ) For estmatg quatles, the above defto of theoretcal quatles s commoly replaced by a lear terpolatg fucto. Ths fucto o oe had esures that all quatles are realzed wth the rage of the emprcal dstrbuto (0 %-quatle equals the mmum of the data, 00 %-quatle equals the mamum of the data). The terpolato o the other had allows a "better guess" of the real quatle f oly few data are gve ad the uderlyg dstrbuto s cotuous. The commoly used computato formula s gve by: where + ( ) α, f + ( ) α q α ( f ) ( ) + f ( + ) ad ( ) : ( ) +. Here () deotes the -th ordered data value ad z deotes the largest teger less or equal to z,.e. 3, 3, 4,9 4. Therefore wth the computato of, the quatle s localzed depedg o the value of α betwee () ad (+). The terpolato betwee these two values s doe accordg to the devato f betwee ad ( +(-) α). Eamples of emprcal CDFs ad emprcal quatles for data smulated from the eample dstrbutos to 3 are gve fgure 5.4. The sold black le represets the emprcal quatles derved by the above formula (from 0 % to 00 %)..

16 6 TS V.. (004-0) Eample, 0 Eample, 0 Eample 3, 0 emprcal F() emprcal F() emprcal F() Fgure 5.4a: Illustrato of emprcal CDFs ad quatles for eamples to 3 Eample, 000 Eample, 000 Eample 3, 000 emprcal F() F() emprcal F() Fgure 5.4b: Illustrato of emprcal CDFs ad quatles for eamples to 3 Note that the above estmato procedure should be appled wth great care for data sets wth oly few data values where the uderlyg dstrbuto s presumably dscrete, sce the estmated quatles also take values dfferg from those cotaed the gve data set. Ths ca also be see from fgure 5.4a the plots for samples wth sample sze Importat dstrbutos I ths clause some of the mportat dstrbutos related to practcal usage telecommucatos are descrbed. Ether the metoed dstrbutos are drectly related to measuremet results or they are ecessary to evaluate these results a secod step. Further relevat dstrbutos may be appeded later. I geeral, dstrbutos are specfed by certa parameters whch descrbe ther ma characterstcs. Commoly, the characterstcs are epressed terms of ther momets,.e. mea value ad stadard devato or varace, respectvely. Wherever possble, the relevat characterstcs are gve as well as eamples of possble use-cases. I geeral, cotuous ad dscrete dstrbutos are dstgushed further o Cotuous dstrbutos A large umber of dfferet cotuous dstrbutos s avalable to descrbe measuremet results a statstcal maer. A overvew s for stace gve by [LAW] or [HART] (see bblography). For practcal purposes the feld of Qualty of Servce (QoS) probg, the dstrbutos descrbed below are probably the most relevat oes Normal dstrbuto The ormal dstrbuto, also called Gaussa dstrbuto (or bell-shaped dstrbuto) s used for may atural processes wheever a symmetrc cotuous dstrbuto seems approprate. (A eample was gve before ad desty fuctos of further ormal dstrbutos are gve fgure 5.5.)

17 7 TS V.. (004-0) Normal dstrbuto Notato X ~ N ( µ, σ ) Parameters µ, σ { } PDF f ( ) ep ( µ ) CDF σ π σ { ( t ) }dt F( ) ep µ σ π Epected value E (X ) µ Varace Var ( X ) σ Remarks Stadard ormal dstrbuto wth µ 0 ad σ, see clause σ The ormal dstrbuto s uquely specfed by ts mea ad stadard devato. For ormally dstrbuted data, about 68 % of the data are realzed wth the terval [µ - σ, µ + σ], 95 % are realzed wth [µ - σ, µ + σ] ad 99,7 % are realzed wth [µ - 3σ, µ + 3σ]. The last terval s also called 6σ-terval whch gave the ame to the popular "S-sgma"-courses Normal PDF Eample wth µ00, σ0 68 % wth [µσ, µ+σ] Normal PDF Eample wth µ0, σ3 95 % wth [µσ, µ + σ] Normal PDF Eample 3 Stadard Normal wth µ0, σ 99.7 % wth / µ 0.3 % outsde [µ3σ, µ + 3σ] µ3σ µ+3σ Fgure 5.5: Desty fuctos of three dfferet ormal dstrbutos Normally (or early ormally) dstrbuted data s foud qute ofte practce, partcular ature, for eample huma or amal body heghts Stadard ormal dstrbuto All ormal dstrbutos or ormally dstrbuted data ca be stadardzed by subtractg the mea ad afterwards dvdg by the stadard devato of the dstrbuto or data resultg a stadard ormal dstrbuto wth mea µ 0 ad stadard devato σ. The verse computato leads back to the orgal dstrbuto or data. Therefore, all ormal dstrbutos may be reduced to the stadard ormal, f the parameters µ ad σ are kow or estmated. Because of ths ad the fact that may statstcal tests are based o the ormal dstrbuto, statstcal tetbooks ofte provde the quatles of the stadard ormal dstrbuto. I partcular, the α -quatle of the stadard ormal dstrbuto s deoted as u α. I eample 3 of fgure 5.5, the desty of the stadard ormal dstrbuto s gve.

18 8 TS V.. (004-0) Stadard ormal dstrbuto Notato X ~ N (0,) Parameters oe PDF f ( ) ep{ } CDF F( ) π ep{ t }dt π Epected value E ( X ) 0 Varace Var ( X ) Remarks Cetral lmt theorem Aother reaso for the frequet use of ormal dstrbutos ( partcular for testg purposes) s gve by the cetral lmt theorem, oe of the most mportat theorems statstcal theory. It states that the mea of equally dstrbuted radom varables wth mea µ ad varace σ approaches a ormal dstrbuto wth mea µ ad varace σ / as becomes larger. Ths holds for arbtrary dstrbutos ad commoly the typcal shape of the ormal dstrbuto s suffcetly reached for 4. For further detals about the cetral lmt theorem see [LAW] or [MOOD] (see bblography). A umber of tools was developed for checkg whether data (or meas) are ormal, amely test procedures lke the well-kow Kolmogorov-Smrov goodess-of-ft test (see clause ) amog others or graphcal tools lke hstograms or QQ-plots. The metoed graphcal tools wll be troduced clause Trasformato to ormalty As has bee see, the ormal dstrbuto s very powerful ad ca be appled may stuatos. Nevertheless, t s ot always approprate, partcular techcal applcatos, where may parameters of terest have o-symmetrc dstrbutos. However, these stuatos t may be possble to trasform the data to ormalty. Ths dea leads for stace to the Log-Normal dstrbuto, whch s ofte assumed for techcal parameters Log-Normal dstrbuto The dstrbuto of a radom varable s sad to be Log-Normal, f the logged radom varable s ormally dstrbuted, whch s deoted by log() ~ N(µ, σ ). Log-Normal dstrbuto Notato X ~ LN( µ, σ ) or log( X ) ~ N( µ, σ ) Parameters µ, σ PDF ( l( ) µ ) ep ( ) f σ π f > 0 σ 0 else CDF F( ) ep{ ( t ) }dt µ σ π σ Epected value E ( ) ep( µ + σ ) Varace Var ( ) ep(µ + σ )(ep( σ ) ) Remarks Log-Normal dstrbutos are skewed ad have heaver upper tals compared to the ormal dstrbuto mplyg a hgher varablty the upper quatles. Desty eamples for dfferet values of µ ad σ are gve fgure 5.6 ad llustrate that the Log-Normal dstrbuto ca take a varety of dfferet shapes.

19 9 TS V.. (004-0) Log-Normal Eample log() s stadard ormal Log-Normal Eample log() s ormal wth µ, σ0.5 Log-Normal Eample 3 log() s ormal wth µ, σ Fgure 5.6: Desty fuctos of Log-Normal dstrbutos Use-case: trasformatos A gve data set ca be checked whether t s dstrbuted accordg to a Log-Normal dstrbuto by computg the log of the data values ad usg oe of the graphcal tools metoed before for verfyg the ormal dstrbuto for the logged data. Emprcal mea ad stadard devato of the trasformed data ca the be used for estmatg the parameters of the dstrbuto, respectvely. Smlarly, other trasformato-based dstrbutos ca be derved from the ormal dstrbuto, for stace for the square-root trasformato ~ IN(µ, σ ) or the recprocal trasformato / ~ IN(µ, σ ). A geeral cocept based o power-trasformatos of was proposed by Bo ad Co (964) Epoetal dstrbuto For modellg arrval processes, ofte the egatve epoetal dstrbuto s used. The relevat parameter for ths dstrbuto s λ whch symbolzes the lfe cycle of a process. Cocerg arrval processes, λ s amed the ter-arrval rate of succeedg evets. Epoetal dstrbuto Notato X ~ Ep(λ) Parameters λ > 0 f ( ) λ ep λ f 0 PDF ( ) CDF F ) ep( λ) Epected value Varace Remarks ( f 0 E { X} λ { X} Var λ Lfe-cycle descrpto, survval fucto: Survval probablty P( X > ) ep( λ)

20 0 TS V.. (004-0) Negatve Epoetal Eample Lambda s Negatve Epoetal Eample Lambda s Negatve Epoetal Eample Lambda s Fgure 5.7: Desty fuctos of egatve epoetal dstrbutos Webull dstrbuto The Webull dstrbuto s a heavy-taled dstrbuto whch meas the dstrbuto s skewed wth a o-eglgble part of the probablty mass the tal. Ths dstrbuto ca be used to descrbe processes whch have a rare frequecy, but whch are ot eglgble due to ther weght. Webull dstrbuto Notato X ~ Webull( α, β ) Parameters α wth α 0, β wth β > 0 PDF β β f ( ) αβ ep( α ) f > 0 CDF β F ( ) ep( α ) f 0 Epected value { } E X α β Γ + β wth Γ Gamma fucto Varace { } Var X α β Γ + Γ + wth Γ Gamma fucto β β Remarks Fatgue of materal Webull (, β ) s a Raylegh dstrbuto wth parameter β. Raylegh s used for descrpto of fadg effects. The Gamma fucto s defed as the tegral fucto Γ( ) ep( t) t dt. Oe mportat relatoshp for the Gamma fucto s gve by ( + ) Γ( ) 0 Γ. For teger values ths relato trasforms to Γ( ) ( )!

21 TS V.. (004-0) Webull Eample Webull data wth α ad β Webull Eample Webull data wth α ad β Webull Eample 3 Webull data wth α3 ad β Fgure 5.8: Desty fuctos of Webull dstrbutos Pareto dstrbuto The Pareto fucto also models a heavy taled dstrbuto. Oe commo use-case of ths dstrbuto s the modellg of packet-oreted data traffc. For eample, the sze of HTTP requests ad reples as well as FTP dowloads ca be descrbed as a Pareto fucto. Pareto dstrbuto Notato X ~ Pareto( c,α ) Parameters c scale ad locato parameter α shape parameter PDF f ( α + ) α ) α c CDF Epected value Varace Remarks ( for > c c F( ) c E{ X} for α > α cα Var{ X} for α > α α ( ) ( ) 4 Pareto Eample Pareto data wth α ad c 4 Pareto Eample Pareto data wth α ad c 4 Pareto Eample 3 Pareto data wth α4 ad c Fgure 5.9: Desty fuctos of Pareto dstrbutos

22 TS V.. (004-0) Etreme dstrbuto (Fsher-Tppett dstrbuto) For modellg etremely seldom evets wth a hgh ad eglgble fluece, the etreme dstrbuto may be approprate. EXAMPLE : EXAMPLE : I servce probg, ths dstrbuto for eample relates to the amout of data whch s trasferred va FTP data coectos. Whereas most of the users geerate traffc the rage of some te or hudred megabytes, at some tme sgle users occur whch lke to trasfer for eample 0 ggabytes oe sesso. Whe modellg the overall FTP data traffc, these users caot be eglected due to ther mmese data volume, but ther occurrece probablty s very low. Cocerg surace cases, sgle cdets whch requre a very hgh facal effort arse whe for eample a eploso elmates a complete factory buldg. Aga, due to the hgh facal effort these cases have be take to accout eve they occur rarely. Etreme dstrbuto X ~ Etreme α, β Notato ( ) Parameters α shape parameter β scale parameter PDF α α f ( ) ep ep ep β β β CDF α F( ) ep ep β Epected value E {} α + βγ wth γ 0, costat of Euler-Maschero Varace Remarks β Π Var{} for α > Etreme Eample Eteme data wth α ad β 0.4 Etreme Eample Etreme data wth α4 ad β 0.4 Etreme Eample 3 Etreme data wth α4 ad β Fgure 5.0: Desty fuctos of etreme dstrbutos 5.6. Testg dstrbutos Statstcal tests are commoly appled to reject a assumpto favour of a alteratve assumpto. Therefore, most tests are based o some kd of measure of devato. Ths may be the devato of data from a model assumpto or from a assumed mea value, a target value ad so o. For computatoal ease, sgle devatos are ofte assumed to be ormally dstrbuted. Based o these cocepts, three mportat testg dstrbutos are troduced the followg, amely the Ch-square-, F- ad Studet t-dstrbutos.

23 3 TS V.. (004-0) Ch-Square dstrbuto wth degrees of freedom If the results of a servce probg s assumed to be the result of a umber of depedet stadard Normal processes, ths dstrbuto provdes a bass for testg agast ths assumpto. For evaluato purposes cocerg the χ dstrbuto, see clause A χ dstrbuto represets a combato of depedet radom varables Z,..., Z where each radom varable s stadard ormal,.e. Z ~ N (0,). The combato s doe accordg to: Z χ ~ The result of ths combato s called a "(cetral) χ dstrbuto wth degrees of freedom". (Cetral) Ch-Square dstrbuto Notato X ~ χ Parameters PDF CDF Radom varable X Z : degrees of freedom Z : depedet stadard ormal radom varables: Z ~ N (0,) Z,..., ( ) f ep for > 0 Γ F ( ) f ( ξ ) dξ No closed soluto avalable Epected value E { X} Varace Var{ X} Remarks Combato of statstcally depedet N (0,) radom varables (stadard ormal) Appromato: F ( ) P X Φ ( ) ChSquare Eample ChSquare data wth ChSquare Eample ChSquare data wth ChSquare Eample 3 ChSquare data wth Fgure 5.: Desty fuctos of Ch-Square dstrbutos

24 4 TS V.. (004-0) Further relatos The refereced gamma fucto s defed as the tegral fucto: 0 Γ( ) ep( t) t dt Addtoal useful relatos accordg to ths fucto are: ad Γ ( + ) Γ( ) Γ( ) ( )! f s a teger value Relato to emprcal varace If the mea value µ s kow, the emprcal varace of ormally dstrbuted radom varables reads µ µ s ( ) sµ epresso: ~ χ. σ. Wth ths pece of formato, a ch-square dstrbuto s gve for the followg Wthout kowledge of µ, the emprcal varace ( ) process. The approprate relato ths case reads ( ) Studet t-dstrbuto s estmates the varace of the s ~ χ. σ If a stadard ormal ad a statstcally depedet ch-square dstrbuto wth degrees of freedom are combed U accordg to X, where Z ~ χ Z (ch-square dstrbuted) ad U ~ N(0,) (stadard ormal dstrbuted), the costructed radom varable X s sad to be t-dstrbuted wth degrees of freedom. Alteratvely, the deomato "Studet t-dstrbuto" ca be used.

25 5 TS V.. (004-0) Notato Parameters Radom varable Studet t-dstrbuto X ~ t U X wth U ~ N(0,), Z~χ Z, depedet. : degrees of freedom PDF + Γ ( ) f + Γ Π CDF F ( ) f ( ξ ) dξ No closed soluto avalable Epected value : E { Z} 0 + Varace 3 : Var { Z} Remarks The PDF s a symmetrc fucto wth symmetry as 0. Addtoal relato for α -quatles t ; α : t ; α t; α 0.4 Studet-t Eample Studet-t data wth 0.4 Studet-t Eample Studet-t data wth Studet-t Eample 3 Studet-t data wth Fgure 5.: Desty fuctos of Studet-t dstrbutos Relato to ormal dstrbuto It may ot be obvous, but t-dstrbutos wth large umber of degrees of freedom may be appromated by a stadard ormal dstrbuto. The stadardzato of ormal varables was covered before: If X ~ N(µ, σ ), the (X-µ)/σ ~ N(0, ). Cosder the case of data assumed to be ormal wth ukow varace. As stated before, the emprcal varace s the related to a ch-square dstrbuto. The emprcal mea ad varace of ormally dstrbuted (N(µ, σ ))) radom varables X, X,..., X are gve by: X X ( ) S X X

26 6 TS V.. (004-0) Wth these relatos, the relato betwee the t-dstrbuto ad the ormal dstrbuted radom varables reads: X µ ~ t. S F dstrbuto The F dstrbuto s a combato of m stadard ormal dstrbuted radom varables Y ad stadard ormal dstrbuted radom varables V whch are combed as descrbed below. Aga, m ad are called "degrees of freedom" of ths dstrbuto. Ths dstrbuto s ofte used for computato ad evaluato purposes, for eample relato wth cofdece tervals for the bomal dstrbuto (Pearso-Clopper formula). I geeral, t compares two types of devatos, for stace f two dfferet models are ftted. Notato Parameters PDF CDF Epected value Varace F dstrbuto ~ X F m, m Y m Radom varable X V m, : degrees of freedom Y,...,Y : depedet radom varables accordg to N (0,) V,...,V : depedet radom varables accordg to N (0,) m m m m + m f ( ) + for > 0 m B, wth B( p, q) Γ( p) Γ( q) Γ( p + q) Eulara beta fucto F ( ) f ( ξ ) dξ No closed soluto avalable > : E { Z} ( ) ( ) ( ) m + m > 4 : Var { Z} m 4 Remarks A F m, related dstrbuto ca be terpreted as the quotet of a χ dstrbuto ad a χ dstrbuto multpled wth. m m

27 7 TS V.. (004-0) F Eample F data wth m 4 ad F Eample F data wth m ad F Eample 3 F data wth m 40 ad Fgure 5.3: Desty fuctos of F dstrbutos Quatles For quatle computato purposes, the followg relatos may be useful: F, ; γ F I geeral, quatle values of ths dstrbuto are tabulated Appromato of quatles, ; γ If the desred quatle value caot be foud tables, the followg appromato may be helpful: If the γ -quatle s wated wth γ the rage 0,5 < γ <, the relato apples where uγ F, ; γ ep ( u a b) u s the γ -quatle of the stadard ormal dstrbuto ( 0,) The symbols a ad b are derved from the followg equatos: N. a d + cd b c + c ( ) u γ 3 6 d d 3

28 8 TS V.. (004-0) Relatos to other dstrbutos Whe the F dstrbuto comes to usage, the followg relatos may ease the hadlg of ths dstrbuto: Relato to t dstrbuto for :, ; ; F γ t +γ. Relato to χ dstrbuto for : F, ; γ χ ; γ. If ad, the dstrbuto smplfes to: F Dscrete dstrbutos, ; γ Dscrete dstrbutos descrbe stuatos where the outcome of measuremets s restrcted to teger values. For eample, the results of servce access tests show ether that servce access s possble (mostly represeted by a logcal "" value) or that t s ot possble (mostly represeted by a logcal "0" value). Depedg o the crcumstaces uder whch such "drawg a ball out of a bo" tests are eecuted, dfferet statstcal dstrbutos apply lke show clauses to Beroull dstrbuto The startg pot of dfferet dscrete dstrbutos s gve by the Beroull dstrbuto. It smply descrbes the probablty p of a postve outcome of a sgle test where oly two states are allowed, geerally a postve oe ad a egatve oe. As soo as more tha oe sgle test s eecuted, further dscrete dstrbuto may be appled as show the followg clauses. Beroull dstrbuto Notato X ~ Beroull( p) Parameters p ( 0,) PDF p p ( ) p 0 f 0 f otherwse CDF 0 F ( ) p f < 0 f 0 < f Epected value E { X} p Varace Var{ X} p ( p) Remarks

29 9 TS V.. (004-0).0 Beroull p s Beroull p s Beroull 3 p s f() f() f() Fgure 5.4: Desty fuctos of Beroull dstrbutos Bomal dstrbuto Wheever the outcome of a test s ether true or false, the bomal dstrbuto ca be appled. I ay case where a "black or whte" terpretato of results s approprate, ths dstrbuto s able to descrbe the measuremet process. Due to ths "yes or o" character, the bomal dstrbuto ca be terpreted as the result of dfferet Beroull tres. Relevat eamples related to servce probg are servce access ssues (e.g. call success rate, SMS sed falure rato, etc.). For a hgh umber of measuremet results, the dstrbuto ca be replaced by the Normal dstrbuto as a frst appromato as show clause Precodto: To determe the CDF of a bomal dstrbuto wth relato to dfferet tests, the sgle evets have to be depedet from each other. Ths meas that the probablty of a successful outcome of dfferet cosecutve tests must ot chage. I cosequece, ths meas a memory-less process where the result of a succeedg test s ot related to the outcome of ts predecessor(s). Notato ( ) Parameters PDF CDF Bomal dstrbuto X ~ B, p Number of tests m Number of successful test outcomes m p Observed probablty of successful outcomes q p Observed probablty of usuccessful outcomes k k P( X k) p ( p) b(, p, k) k wth k 0,,,..., k0 k k P ( X k0 ) p ( p) k 0 k wth k 0,,,..., ad k0 k E X Epected value { } p Varace Var{ X} p q p ( p) Remarks Related to F dstrbuto

30 30 TS V.. (004-0) f() Bomal s 0, p s f() Bomal s 0, p s f() Bomal 3 s 0, p s Fgure 5.5: Desty fuctos of bomal dstrbutos For computato purposes, the followg relato betwee the bomal dstrbuto ad the F dstrbuto may be useful: P( X < ) P F + I ths formula, F represets a F dstrbuted radom varable wth ( + ), ( ) degrees of freedom Geometrc dstrbuto The geometrc dstrbuto typcally descrbes the followg stuato: A umber of Beroull trals s eecuted cosecutvely. Each of these trals has a success probablty p. By use of the geometrcal dstrbuto, oe ca determe the probablty of a successful outcome of a Beroull tral after usuccessful outcomes. Scearos where ths computato may be of terest are for eample the occurrece of the frst success after falures, or related to servce probg, the umber of faled servce access attempts before the frst successful attempt. p p Geometrc dstrbuto X ~ G p Notato ( ) Parameters p ( 0,) PDF CDF Epected value Varace Remarks p p( ) ( p) f { 0,,...} 0 F( ) 0 E Var ( ) p { X} otherwse + p p { X} f 0 otherwse p p

31 3 TS V.. (004-0) 0.8 Geometrc p s Geometrc p s Geometrc 3 p s f() 0.4 f() 0.4 f() Fgure 5.6: Desty fuctos of geometrc dstrbutos Posso dstrbuto The Posso dstrbuto s also called "dstrbuto of rare evets". Geerally, ths dstrbuto relates to the umber of evets wth a certa tme of perod uder the precodto that the evets occur at a costat rate λ. The Posso dstrbuto ofte s used to descrbe call arrvals a trasmsso system, especally the curret umber of processed servce attempts a system. Posso dstrbuto X ~ Po λ Notato ( ) Parameters PDF CDF λ λ P( X k) ep( λ) k! wth k 0,,,..., k λ P( X k) ep( λ)! 0 wth k 0,,,..., Epected value E { X} λ Varace Var { X} λ Remarks Related to k k χ dstrbuto f() Posso Lambda s f() Posso Lambda s f() Posso 3 Lambda s Fgure 5.7: Desty fuctos of Posso dstrbutos

32 3 TS V.. (004-0) For computato purposes, the followg relato betwee the Posso dstrbuto ad the χ dstrbuto may be useful: P ( X ) P( χ λ) I ths formula, χ represets a χ dstrbuted radom varable Trastos betwee dstrbutos ad approprate appromatos Depedg o the umber of avalable measuremet results, dfferet dstrbutos ca be appled to hadle the results. I ths clause, some useful trastos betwee commo dstrbutos ad ther requred codtos are dscussed From bomal to Posso dstrbuto The bomal dstrbuto ca be appromated by the Posso dstrbuto f: the probablty p s small (rule of thumb: p < 0, ); ad the umber of eecuted test cases s hgh eough (rule of thumb: > 30 ). The appromato of a bomal dstrbuted quatty by a Posso dstrbuto s gve by: where the Posso dstrbuto parameter λ s gve by: λ P( X k) ep( λ) k! k λ p From bomal to Normal dstrbuto If a bomal dstrbuto fulfls the rule of thumb: p q 9 the t ca be appromated by the Normal dstrbuto: B(, p) N( p, p q) The appromato detal reads: X p P ( X ) Φ p q Especally for smaller umbers of the followg appromato may be more favourable: P( X ) Φ p + 0,5 Φ p q p 0,5 p q From Posso to Normal dstrbuto Accordg to the Posso lmt theorem, the Posso dstrbuto ca be appromated to the Normal dstrbuto f the dstrbuto parameter λ fulfls the followg relato: λ p 9 whch s qute smlar to the trasto from bomal to Normal dstrbuto.