Chapter One Introduction

Transcription

1 Chapter Oe Itroducto Statstcs: Def. (): The term statstcs refers to the procedures ad formulas used to aalze data ad to the umercal aswers obtaed from statstcal procedures. Statstcal procedures are used to orgaze, summarze, ad commucate data ad to draw coclusos about what the data dcate. The goal learg statstcs s to kow whe to perform a partcular procedure ad how to terpret the aswer. Def. (): Statstcs s cocered wth scetfc methods for collectg, orgazg, summarzg, presetg, ad aalzg data, as well as drawg vald coclusos ad makg reasoable decsos o the bass of such aalss. Populato: A populato represets the etre group of data cocerg characterstcs of a group of dvduals or objects. Sample: It s a specfed umber of measuremets or data draw from the populato. If a sample s represetatve of a populato, mportat coclusos about the populato ca ofte be ferred from aalss of a sample. Epermetal ut: t s the dvdual or object o whch a varable s measured. Whe a varable s actuall measured o a set of epermetal uts, a set of measuremets or data result. Descrptve Statstcs: The brach of statstcs cocered wth the procedures used to summarze ad descrbe the mportat characterstcs of a set of measuremets.

2 Statstcal ferece: It deals wth codtos uder whch mportat coclusos ca be ferred from aalss of the sample. هو استخدام المعلومات المتوفرة ف الع نة لعمل تنبؤ او اتخاذ قرار او تعم م النتائج على المجتمع الذي سحبت منه الع نة. Varable: s a characterstc that chages or vares over tme ad/or for dfferet dvduals or objects uder cosderato. Qualtatve varables: measure a qualt or characterstc o each epermetal ut. Qualtatve varables produce data that ca be categorzed accordg to smlartes or dffereces kd; the are ofte called categorcal data. Quattatve varables: ofte represeted b the letter produce umercal data measured o each epermetal ut, such as heght, weght, dstace, tme, volume, age, umber of faml members, etc. We defe two tpes of quattatve varables: A Dscrete varable: ca assume ol a fte or coutable umber of values. A cotuous varable: ca assume values at a pot alog a le terval. Summato Notato: s used to deote the sum of all values of from =,,,. The Greek captal letter deote the sum ad subscrpt deote the values,, 3,,. Thus Ad 3...

3 3 Some rules for otato:. c c c c c c tmes... where c s a costat. Eample: Note that: c a c a Eample: c c Eample: z z

4 4 Eamples b a b a a a Product otato: log... log log log log Note that:

5 5 Collecto of data Oe of the ma fuctos of statstcs s to provde formato whch wll help makg decsos. A statstcal data ca be classfed uder two categores depedg o the sources utlzed. These categores are: - Prmar data - Secodar data Prmar data s the oe, whch s collected b the vestgator hmself for the purpose of a specfc qur or stud. Such data s orgal character ad s geerated b surve coducted b dvduals or research sttuto or a orgazato. It ca be collected b the followg methods - Drect persoal tervews The vestgator persoall meets the persos from whom the formatos are collected, ad asks questos to gather the ecessar formatos. - Idrect oral tervews The vestgator cotact wtesses or eghbors or freds or some others who are capable of supplg the ecessar formato. 3- Iformato from correspodets B ths method, formatos come to ewspapers ad some departmets of govermet. 4- Maled questoare method Uder ths method a lst of questos called a questoare s prepared ad s set to all formats b post. A coverg letter accompag the questoare eplas the purpose of the vestgato ad the mportace of correct formatos provded. 5- Schedules set through eumerators.

6 6 A schedule s flled b the tervewers a face-to face stuato wth the format. Secodar data are those data whch have bee alread collected ad aalzed b some earler agec for ts ow use, ad later the same data are used b a dfferet agec. The sources of secodar data ca broadl be classfed uder two heads:. Publshed sources: such as reports ad offcal publcatos, sem offcal publcatos, prvate publcatos.. Upublshed sources: such as records mataed b varous govermet ad prvate offces, studes made b research sttutos, etc. Tabulato of data It s the process of summarzg classfed or grouped data the form of a table so that t s easl uderstood, ad a vestgator s quckl able to locate the desred formato. Thus, a statstcal table makes t possble for the vestgator to preset a huge mass of data a detaled ad orderl form. It facltates comparso ad ofte reveals certa patters data whch are otherwse ot obvous.

7 7 Frequec Dstrbutos Chapter Two Descrptve Statstcs Frequec dstrbuto s smpl a table whch the data are grouped to classes ad shows the umber of cases fall each class. Data orgazed a frequec dstrbuto are called grouped data; otherwse the are called ugrouped data. Raw data: The statstcal data whch have ot bee orgazed umercall are geerall called raw data. Arra: Is a arragemet of raw umercal data ascedg or descedg order of magtude. A data set havg a relatvel small umber of dstct values ca be preseted a frequec table. Eample: A studet receved the followg grades o the 0 quzzes he took durg a semester. 6, 7, 6, 8, 5, 7, 6, 9, 0, 6 Grades Tall Frequec Total 0

8 8 Data from a frequec table ca be graphcall represeted b a le graph that plots the dstct data values o the horzotal as ad dcates ther frequeces b the heghts of vertcal les. Whe the les a le graph are gve added thckess, the graph s called a bar graph. Relatve Frequec: For a data set cosstg of values, ad f s the frequec of a partcular value, the the rato f / s ts relatve frequec. The relatve frequeces ca be represeted graphcall b relatve frequec graphs. The look lke the correspodg graphs of the absolute frequeces ecept that the labels o the vertcal as are the proporto of the data pots. Eample: Below s a frequec table for a data set cosstg of the starg earl salares (to the earest thousad dollars) of 4 recetl graduated studets wth B.S. degrees electrcal egeerg. Startg Frequec Relatve Percet salar frequec /4 9.5% 48 /4.4% /4 7.% /4.9% 5 8 8/4 9.% 5 0 0/4 3.8% /4.9% 56 /4 4.8% /4 7.% 60 /4.4% Total 4 00%

9 9 Pe Charts: A pe chart s ofte used to dcate relatve frequeces whe the data are ot umercal ature. A crcle s costructed ad the slced to dfferet sectors; oe for each dstct tpe of data value. The area of a certa sector s equal to the total area of the crcle multpled b the relatve frequec of the data value. Eample: The followg data relate to the dfferet tpes of cacers affectg a 00 patets erolled at a clc specalzg cacer. Tpe of cacer Frequec Relatve frequec Percet Agle Lug 4. %. 360=75.6º Breast % 90.0º Colo 3.6 6% 57.6º Prostate % 99º Melaoma % 6.º Bladder.06 6%.6º Total % 360º

10 0 I the case of large data sets, t s useful to dvde the values to groupgs, or class tervals. The umber of class tervals chose s tpcall betwee 5 ad 0. It s commo, although ot essetal, to choose class tervals of equal sze. The edpots of a class terval are called the class boudares. We wll adopt the left-ed cluso coveto whch stpulates that the class terval 0-30 cotas all values that are both greater tha or equal to 0 ad less tha 30. The class sze or wdth s the dfferece betwee the lower ad upper class boudares. The class mark or mdpot s the ceter of the class terval. Geeral Rules for formg frequec dstrbutos: - Determe the largest ad smallest umbers the raw data ad fd the rage Rage = largest umber smallest umber - Dvde the rage to a coveet umber of classes havg the same sze f possble. 3- Determe the umber of observatos fallg to each class. Eample: The data below represet the weghts kg of 50 college studets. Costruct a frequec dstrbuto for the data Sze of class terval = Rage / Number of classes = (64 3) / 7 = 3 / 7 = Choosg 7 class tervals, the sze of each class s 5. The requred frequec dstrbuto s prepared as gve below:

11 Class terval Tall marks Frequec Relatve frequec Total Cumulatve frequec Dstrbutos: A less tha cumulatve frequec dstrbuto s the total frequec of all values less tha the upper class boudar of a gve class terval. Weghts Cumulatve Frequec Less tha Less tha Less tha Less tha Less tha Less tha Less tha Cumulatve Relatve Frequec A cumulatve frequec dstrbuto of all values greater tha or equal to the lower class boudar of each class terval s called a More tha cumulatve frequec dstrbuto. Weghts Cumulatve Frequec 30 ad above ad above ad above ad above ad above ad above ad above 4.08 Cumulatve Relatve Frequec

12 Hstograms ad Frequec Polgos: A hstogram or frequec hstogram cossts of a set of rectagles havg: - Bases o the horzotal as wth ceters at the class marks, ad legths equal to the class sze. - Areas proportoal to class frequeces. If the class tervals all have equal sze, the heghts of the rectagles are proportoal to the class frequeces ad t s the customar to take the heghts umercall equal to the class frequeces. If class tervals do ot have equal sze, these heghts must be adjusted. If the vertcal as showed the relatve frequeces the result would be called a relatve frequec hstogram A frequec polgo s a le graph of class frequec plotted agast class mark. It ca be obtaed b coectg mdpots to the tops of the rectagles the hstogram. It s customar to add etesos to the et lower ad hgher class marks whch have correspodg class frequec of zero. I such case the sum of areas of the rectagles the hstogram equals the total area bouded b the frequec polgo ad the -as.

13 A frequec curve s a smoothed frequec polgo

14 4 A cumulatve frequec polgo s called a ogve H.W. - Followg s the umber of emergec calls made 35 cosecutve das b a ambulace server compa. Costruct a frequec dstrbuto ad sketch the relatve frequec hstogram ad polgo I the U.S. ar polluto data are reported for 57 ctes. Costruct a frequec dstrbuto ad sketch the less tha ad more tha cumulatve frequec polgos

15 5

16 Chapter Three Averages ad Measures of Cetral Tedec A average s a value whch s tpcal or represetatve of a set of data. Such tpcal values ted to le cetrall whe the data are arraged accordg to magtude. Therefore, averages are also called measures of cetral tedec.. Arthmetc Mea: Ths s the most commo tpe of average, brefl called as the "Mea". It s the sum of the observatos dvded b ther umber.... Eample: Cosder the followg 0 observatos ordered from the smallest to the largest, each oe represetg the lfetme hours of a certa tpe of lamp. 6, 63, 666, 744, 883, 898, 964, 970, 983, 003, 06, 0, 09, 058, 085, 088,, 35, 97,

17 Grouped data: Whe the frequec of some of the observatos s greater tha, computato ma be smplfed b usg frequec groupg. If f s the frequec of a value, the the mea for grouped data ca be wrtte as: Eample: f f, where f =. Fd the arthmetc mea for the followg data.. Costruct a frequec dstrbuto for the data ad fd the arthmetc mea for the grouped data. 37, 43, 4, 46, 37, 44, 38, 39, 37, 4, 38, 45, 38, 48, Rage = = Class wdth = /4 = Lower class lmt = 37 Upper class lmt = 39 Class f f d f d

18 3 f f Codg Method: A further savg effort s effected b reducg all observatos b a costat value or b dvdg them b a factor such bas 0, ths s kow as codg. Hece f A s a arbtrar costat, d = A, The: d A for ugrouped data Let A = 37, the d becomes: 0, 6, 5, 9, 0, 7,,, 0, 5,, 8,,, 6 A d For grouped data: A where A s a arbtrar class mark. Let A = 38 A f f d f f d Weghted Mea: Sometmes we assocate wth the umbers,,, certa weghts w, w,, w depedg o the sgfcace or mportace attached to the umbers. I ths case,

19 4 w w w w w w w... w... w Eample: Suppose that a semester ou receved a grade of 70 a 3-hour course, 84 a 4-hour course, ad 90 a 5-hour course. Compute our weghted average semester's grade. w w w w w w w w w Propertes of the Arthmetc Mea:. The algebrac sum of devatos of a set of umbers from ther arthmetc mea s zero:.e.; proof : 0 0. The sum of squares of devatos of a set of umbers from s the mmum:.e.; a, where a s a umber

20 5 Proof: ) ( a a a a a a a a a a a a a a a a a a a a a a a a Eample: ( ) ( 5) ( 5) = + 5 = For a costat a, f a The a Proof: a a a

21 6 Eample: Let a = For a costat a, f a The a Proof: a a a Eample: Let a = If z The z Proof: z z Eample: Let z = + z

22 7 z z Or z The Geometrc Mea: It s defed as the th root of the product of observatos. G It s of terest egeerg calculatos, ad s used whe dealg wth observatos each of whch bears a appromatel costat rato to the precedg oe. For eample averagg rates of growth (crease or decrease) of a statstcal populato. I practce, G s computed b logarthms: log G log Eample: log log log 3... log log Fd the geometrc mea of the umbers: 3, 5, 6, 6, 7, 0, ad 7 G or log G 7 log3 log5 log 6 log 6 log 7 log0 log

23 8 G At log G G alwas Grouped data: For grouped data the geometrc mea s calculated as follows: G Or log G f f k. f log f... log f f k log k where f fk.... k f log... f k k f log k Where:,,, k are class marks f, f,, f k are class frequeces Eample: Fd the geometrc mea for the followg frequec dstrbuto: Class f log f log f d f d

24 9 log G k f log G = 68.9 f Or let A = Meda: d = A The meda of a set of observatos s the mddle observato whe the are raked or arraged order of magtude. That s, the meda for ugrouped data s the value of the th tem the data arra, f s odd. If s eve the meda s take as oe-half the sum of the two mddle values Eample :,. Fd the meda of 0,, 0, 7,, 6, ad 5 0,,, 5, 6, 7, 0 Meda = 5 7 4

25 0 Eample : Fd the meda of 8, 6, 9, 5, 5,, 0, ad 7 5, 5, 6, 7, 8, 9, 0, 4, Meda 7.5 Grouped data: For grouped data, the meda s gve b the formula: / Meda L f m F m c where L = lower lmt of the meda class (the class that cotas the mddle tem of the dstrbuto). = umber of observatos = f F m- = sum of the frequeces up to but ot cludg the meda class. f m = frequec of the meda class. c = class sze. Geometrcall, the meda s the value of that dvdes the hstogram or a frequec polgo or curve to two equal areas. Because the meda s a postoal value, t s less affected b etreme values tha the mea. Ths propert of the meda makes t some cases a useful measure of cetral tedec.

26 Eample: Fd the meda for the followg frequec dstrbuto: Class f F d = 67.6 f d The meda class s the frst class that has cumulatve frequec greater tha or equal /. The meda class s: Meda / F m L c f m / G Mode 67. or 67.5

27 Eample: The hourl wages of fve emploees a offce are:.5, 3.96, 3.8, 9.0, 3.75 Fd:. Meda. Mea. Arrage a arra.5, 3.8, 3.75, 3.96, 9.0 Meda = $ Mea $ Mode: Is the value of the observato whch occurs most frequetl,.e., t s the most commo value. The mode ma ot est ad f t does est t ma ot be uque. A dstrbuto havg ol oe mode s called umodal. Eample: The set,, 5, 7, 9, 9, 9, 0, 0,,, 8 has mode 9. The set 3, 5, 8, 0,, 5, 6, 0 has o mode. The set, 3, 4, 4, 4, 5, 5, 7, 7, 7, 9 has modes 4 ad 7 ad s called bmodal. Grouped data: For grouped data, the mode wll be the value of correspodg to the mamum pot o the curve. It ma represet the class mdpot of the modal class, or t ca be obtaed b the formula:

28 3 Mode where L c L: lower lmt of modal class (class correspodg to the hghest frequec). : dfferece betwee modal class ad ts predecessor. : dfferece betwee modal class ad ts successor. c: modal class sze. Eample: Followg s the dstrbuto of the amout of tme spet the eercse room of a health club b a sample of 75 patros. No. mutes f Mode L 8 c Or: Mode 37

29 4 Comparso betwee Mea, Meda ad Mode: For umodal frequec curves whch are moderatel skewed, we have the emprcal relato: Mea Mode = 3(Mea Meda) Or Mea Meda = /3(Mea Mode) Whe the three averages do ot cocde, the frequec dstrbuto curve s sad to be skewed. It s the degree of asmmetr, ad geeral ts value must fall betwee -3 ad 3. Mea Mode Sk S. D 3 Mea Meda S. D The above two measures are called, respectvel, Pearso s frst ad secod coeffcets of skewess. It ma be used to compare the skewess of dfferet dstrbutos. For smmetrcal dstrbutos the value of skewess s zero. Eample: Fd the coeffcet of skewess for the dstrbuto whch has mea = 56.7, the meda = 56. ad stadard devato = 5.4. Sk 3 Mea Meda S. D O the bass of ths result we ca sa that the dstrbuto s earl smmetrcal.

30 - - Chapter s Correlato ad Regresso Correlato: Correlato aalss measures the degree of relatoshp betwee the varables. Whe ol two varables are volved we speak of smple correlato. Whe more tha two varables are volved we speak of multple correlato. Lear Correlato: A frst step s the collecto of data. Suppose ad deote respectvel the heghts ad weghts of adult males. A et step s to plot the pots (, ), (, ),, (, ). The resultg set of pots s called a scatter dagram. If all pots ths scatter dagram seem to le ear a le we sa that the correlato s lear. If teds to crease as creases the correlato s called postve or drect correlato. If teds to decrease as creases the correlato s called egatve or verse correlato. If all pots seem to le ear some curve, the correlato s called o-lear. If there s o relatoshp dcated betwee the varables, we sa that there s o correlato betwee them or the are ucorrelated.

31 - - Smple correlato coeffcet: If a lear relatoshp betwee two varables s assumed, the quatt r called the coeffcet of correlato s gve b: s s r ), cov( s s s Where cov(, ) = s s called the covarace of ad. The covarace measures the etet to whch two varables "var together". The formula for the sample covarace s: ), cov( s, s

32 - 3 - The formula of r becomes: r Ths ca be smplfed to: r Or r The quatt r vares betwee - ad +. The sg are used for postve ad egatve correlato respectvel. Note that r s dmesoless quatt. I order to prove that r, we beg wth ( ) ( ) ( ) ( )( ) Or + ( ) r 0 showg that r To show that r -, we start wth ( ) Ad use the same argumet gve above. Hece, - r

33 - 4 - Eample: Fd the coeffcet of lear correlato betwee the varables ad. ( )( ) ( ) ( ) , 56, 40, 54, 56, 364,( ) = 3, ( ) = 56, ( )( ) = 84 r Or r [( )( )] Eample: O the bass of the followg data, determe whether there s a relatoshp betwee the tme, mutes, t takes a secretar to complete a certa form the morg ad the late afteroo.

34 - 5 - Morg() Afteroo() , r 86.7, 88.8, 77.35, 89.34, Rak Correlato: Istead of usg precse values of the varables, or whe such precso s ot avalable, the data ma be raked order of sze, mportace, etc., usg the umbers,,,. If two varables are raked such a maer the coeffcet of rak correlato also called Spearma's rak correlato coeffcet s: r rak = 6 ( d ) where d = dffereces betwee raks of correspodg values f ad. = umber of pars of values (, ) the data. Eample: The followg are scores whch studets obtaed the mdterm ad fal eamatos a course statstcs.

35 - 6 - R R d d r rak = 6 ( d = ) Multple Correlato: The degree of relatoshp estg betwee three or more varables s called multple correlato. If R.3 s the coeffcet of multple correlato of o ad 3. The R.3 r r 3 r r r r If R.3 s the coeffcet of multple correlato of o ad 3. The R.3 r r 3 r r r 3 3 r3

36 - 7 - If R 3. s the coeffcet of multple correlato of 3 o ad. The R 3. r 3 r 3 r r r 3 3 r A coeffcet of multple correlato les betwee 0 ad. The closer t s to the better s the lear relatoshp betwee the varables. The closer t s to 0 the worse s the lear relatoshp. If the coeffcet of multple correlato s, the correlato s called perfect. Eample: For the data the followg table: - Compute r, r 3 ad r 3. - Compute R.3, R.3 ad R , 643, 06, 4839, r, 40830, 6796, , r 3 = =

37 - 8 - r = R.3 r r 3 r r r r = R.3 r r 3 r r r 3 3 r = R 3. r 3 r 3 r r r 3 3 r = Note that the coeffcet of multple correlato, such as R.3 s larger tha ether of the coeffcets r or r 3. Ths s alwas true, sce b takg to accout addtoal varables we should arrve at a better relatoshp betwee the varables. Partal Correlato: It s ofte mportat to measure the correlato betwee two varables whe all other varables volved are kept costat,.e. whe the effects of all other varables are removed. Ths ca be obtaed b defg a coeffcet of partal correlato.

38 - 9 - If we deote b r.3 the coeffcet of partal correlato betwee ad keepg 3 costat, we fd that: r.3 r r r r 3 3 r 3 3 Eample: Compute the coeffcets of lear partal correlato ) r.3 ) r 3. ) r 3. for the data of the prevous eample. r r r r r 3 r r r r r r r r r r r r r It follows that for costat 3 the correlato coeffcet betwee ad s For costat the correlato coeffcet betwee ad 3 s ol 0.33, ad fall for costat the correlato coeffcet betwee ad 3 s H.W. The followg table reports salar data for = 4 radoml sampled sstems aalsts wth ther ears of eperece ad ears of post secodar educato.

39 - -. Compute the coeffcets of multple correlato R.3, R.3 ad R 3... Compute the coeffcets of partal correlato r.3, r 3., ad r 3.. Aual salar$ ears of eperece ears of postsecodar educato Regresso It s frequetl desrable to epress the relatoshp betwee two (or more) varables mathematcal form b determg a equato coectg the varables. I ma stuatos, there s a sgle respose varable also called depedet varable, whch depeds o the value of a set

40 - - of depedet varables,,, r. The smplest tpe of relatoshp s a lear relatoshp. That s, for some costats β 0, β,, β r the equato = β 0 + β + β + + β r r + e s called a lear regresso equato. It descrbes the regresso of o the set of depedet varables,,, r. The quattes β 0, β,, β r are called the regresso coeffcets ad e represetg the radom error. A regresso equato cotag a sgle depedet varable s called a smple regresso equato, whereas oe cotag ma depedet varables s called multple regresso equato. Smple lear regresso: A scatter dagram s a graph whch each plotted pot represets a observed par of values for the depedet ad the depedet varables. If all the pots the scatter dagram seem to le ear a le we sa that a lear relatoshp ests betwee the varables. The smple lear regresso model ca be epressed as e Where : value of the depedet varable the th observato. α: frst parameter of the regresso equato, whch dcates the value of, whe = 0. β: secod parameter of the regresso equato, whch dcates the slope of the regresso le.

41 - - : the specfed value of the depedet varable the th observato. e : radom samplg error the th observato. The parameters α ad β the regresso model are estmated b the values ˆ ad ˆ that are based o the sample data. The lear regresso equato based o the sample data s called a regresso le of o sce s estmated from. ˆ ˆ ˆ Least Squares Estmators of the Regresso Parameters: To determe estmators of α ad β we reaso as follows: If ˆ s the estmator of α ad ˆ s the estmator of β the the estmated regresso le possble ft to the gve data. ˆ ˆ ˆ provdes the best Sce e s the dfferece betwee the actual respose ad the estmated respose ŷ, that s: e = - ŷ The least squares estmates of the regresso coeffcets are the values of ˆ ad ˆ for whch the quatt Q ˆ e ˆ, s mmzed Dfferetatg partall wth respect to ˆ ad ˆ, ad equatg these partal dervatves to zero we obta: Q ˆ ˆ ˆ 0

42 - 3-0 ˆ ˆ ˆ Q whch eld the followg ormal equatos: ˆ ˆ ˆ ˆ If we let ad, we ca wrte the frst ormal equato as ˆ ˆ Substtutg ˆ to the secod ormal equato elds ˆ ˆ ˆ ˆ Or ˆ So ˆ Or ˆ Resduals:

43 - 4 - The dfferece betwee the observed value of ad the ftted value ŷ s called the resdual for that observato ad s deoted b e, that s e ˆ The set of resduals for the sample data serve as the bass for calculatg the stadard error of estmate. The Stadard Error of Estmate: It s the codtoal stadard devato of the depedet varable gve a value of the depedet varable. s ˆ e Note that the umerator s the sum of the squares of the resduals. A alteratve computatoal formula whch does ot requre determato of each ftted value s s ˆ ˆ Eample: Cosder the followg 0 data pars (, ), =,, 0, o the umber of hours whch 0 persos studed for Frech test ad ther scores o the test.. Fd the equato of the least squares le that appromate the regresso of the test scores o the umber of hours studed.

44 Predct the average test score of a perso who studed 4 hours for the test. Hours studed() Test scores() Plottg these data, we get the mpresso that a straght le provdes a reasoabl good ft.. 0, 00, 376, 564, ˆ ˆ 6945 ˆ ˆ ˆ Or ˆ 70 Note: The arthmetc sg assocated wth β the regresso equato, dcate the drecto of the relatoshp betwee ad (postve = drect, egatve= verse).

45 - 6 - H.W. The followg data are measuremets of the relatve humdt a storage locato ad the mosture cotet of a sample of raw materal take over 5 das.. Ft a least squares le ad use t to predct the value of mosture cotet whe the relatve humdt s 65.. Fd ŷ values correspodg to. Relatve humdt % Mosture cotet% Relatoshp betwee Regresso ad Correlato Coeffcets: The coeffcet of the regresso le of Y o X or Y gve X ca be wrtte as: ˆ S S S S r Or r S S r S S S S S S S S ˆ ˆ ˆ ˆ

46 - 7 - Smlarl the coeffcet of the regresso le of X o Y or X gve Y ca be wrtte as: r S S ˆ The Coeffcet of Determato: It s the square of r ad s deoted b R used for measurg the proporto of varace the depedet varable that s statstcall eplaed b the regresso equato. R les betwee 0 ad. The total varato of Y s the sum of the eplaed varato b the regresso equato ad the ueplaed varato. ˆ ˆ Total sum of Squares = Eplaed sum of squares + Ueplaed sum of squares ) ( ) ˆ ( ) ( ) ˆ ( ˆ ˆ S S R Or, ˆ R For computatoal purposes, the followg formula s coveet: ˆ ˆ R Eample:

47 - 8 - Gve that: 3.5, 6, r = 0.905, S =.333, S =., fd:. The regresso equato of Y o X ˆ ˆ S S r ˆ ˆ The regresso equato of X o Y ˆ ˆ S.333 r S. ˆ ˆ The coeffcet of determato R = (r ) = (0.905) = 0.8 Or, ˆ R Or, R ˆ S S S S

48 Chapter Four Measures of Varato The degree to whch umercal data ted to spread about a average value s called the varato or dsperso of the data. There are several measures of scatter or dsperso, the most commo beg the followg:. The Rage: It s the dfferece betwee the two etremes of the data. R = X - X The rage s eas to calculate ad eas to uderstad, but t tells us othg about the dsperso of data whch fall betwee the to etremes. The followg sets of data Set: Set : Set3 : has a rage of 7-5 =, but the dsperso s qut dfferet each case. Nevertheless, the rage s a ver useful measure of varato whe the sample sze s qut small. It s used wdel dustral qualt cotrol.. Average Devato or Mea Devato: It s the mea of the absolute values of devotos: M. D The use of absolute values s ecessar because the algebrac sum of devatos from the mea s alwas zero.

49 For grouped data: M. D f where s the th class mdpot, = f Eample: Fd M.D for of umbers 3, 5, 6, 8, 9, = M.D = 4 = = = Eample: Fd M.D for the followg frequec dstrbuto: Class f f f f f 45.6 M. D Stadard Devato:

50 3 It s the root mea square of the devato from the mea, ad s deoted b s for the sample stadard devato ad σ for the populato stadard devato. s For grouped data the stadard devato ca be wrtte as: f s Sometmes the stadard devato for the data of a sample s defed wth - replacg the deomator; the resultg value represets a better estmate of the stadard devato of a populato from whch the sample s take. For large values of ( > 30) there s practcall o dfferece betwee the two formulas. A more coveet form ca be obtaed usg the algebrac dett:

51 4 s for ugrouped data Ad, f f s for grouped data The Codg Method: If d = A are the devatos of from some arbtrar costat A, the d d s for ugrouped data Ad, d f d f s for grouped data Eample: Fd the stadard devato for the followg data: d = 5 d

52 s Or, s Or, d d s 4. Varace: Is defed as the square of the stadard devato ad s deoted b s for the sample varace ad σ for the populato varace. For ugrouped data: s Or, s Or, d d s Ad, for grouped data:

53 6 f s Or, f f s Or, d f d f s Eample: Fd the varace for the followg data: d = 0 d s s

54 7 Or, s d d Eample: Fd the stadard devato ad the varace for the followg frequec dstrbuto usg the three formulas: 44.5 Class f f ( ) f f f s f s 4 Let A = 3 f f Class f d = -3 f d f d

55 s 4 f d fd s H.W: For the followg frequec dstrbuto, fd: M.D, S.D, ad varace usg the three formulas. Class: f : Propertes of the Stadard Devato ad the Varace:. If = ± a, the s s Proof: s a a a a s where a. If = a, the s a s Proof:

56 9 s a a where a a a a s Eample: = +5 z = v = 4 w = / s s 5.8 ad sz 5.8, s v ad s Coeffcet of Dsperso (varato): It s a measure of relatve dsperso, ad s deoted b v: w 4

57 0 v s 00% It s geerall epressed as a percetage. Note that the coeffcet of varato s depedet of uts. For ths reaso t s useful comparg dstrbutos where uts ma be dfferet. A dsadvatage of the coeffcet of varato s that t fals to be useful f s close to zero. Eample: A maufacturer of televso tubes has two tpes of tubes, A ad B. The tubes have respectve mea lfetmes A =495 hours ad =875hours ad stadard devatos S A =80hours ad S B =30 hours. Whch tube has greater relatve varato? B v A s A A 80 00% 00% 8.7% 495 v B s B B 30 00% 00% 6.5% 875 Tube A has greater relatve varato.

58 Momets Chapter Fve If,,, are the values assumed b the varable, the the quatt s called the rth momet about the arthmetc mea. Momets ca be defed as the arthmetc mea of varous powers of devatos take from the mea of a dstrbuto. These momets are kow as cetral momets. If r =, m = 0. If r =, m = s, the varace. The rth momet about a org A s defed as: If A = 0, m r, s ofte called the rth momet about zero. The frst momet about zero wth r =, s the arthmetc mea. The momets about a org are kow as raw momets. Eample:. Fd the frst four cetral momets for the followg data:, 3, 7, 8, ad 0.. Fd the frst four raw momets about org zero.

59 3. Also Fd the frst four raw momets about org A= 4. Momets for grouped data If,,, k occur wth frequeces f, f,, f k, the: Relatoshp betwee momets: where

60 3 I geeral: Eample: Prove that: Let the ad = H.W.:. Prove that:. From the data gve below, fd the frst four momets about a arbtrar org, the calculate the frst four momets about the mea b applg the above relatoshps Class Frequec Skewess:

61 4 We stud skewess to have a dea about the shape of the curve of a gve data. It s the degree of asmmetr of a dstrbuto. For a smmetrc dstrbuto Mea = Meda = Mode, otherwse t s called a skewed dstrbuto, ad such a dstrbuto could ether be postvel skewed or egatvel skewed. A measure of asmmetr s suppled b the dfferece (mea mode). Ths ca be dmesoless o dvso b a measure of dsperso such as the stadard devato, leadg to the followg defto: Or: The above two measures are called, respectvel, Pearso s frst ad secod coeffcets of skewess. A mportat measure of skewess uses the thrd momet about the mea epressed dmesoless form s called momet coeffcet of skewess.

62 5 Kurtoss: Measure of kurtoss tells as the etet to whch a dstrbuto s more peaked or more flat topped tha the ormal curve, whch s smmetrcal ad bell-shaped. It s the degree of peskess of a dstrbuto, usuall take relatve to a ormal dstrbuto. A dstrbuto havg relatvel hgh peak s called leptokurtc, whle a curve whch s flat-topped s called platkurtc. The ormal curve s called mesokurtc. Oe measure of kurtoss uses forth momet about the mea epressed dmesoless form s called momet coeffcet of kurtoss. For the ormal dstrbuto β = 3. Sometmes kurtoss s measured as the dfferece (β 3), for ths reaso the kurtoss for a leptokurtc dstrbuto s postve (β > 3 or β 3 > 0), ad s egatve for a platkurtc dstrbuto (β < 3 or β 3 < 0), ad zero for the ormal dstrbuto. Eample: Fd the momet coeffcet of skewess ad kurtoss for the followg table. Class mark

63 6 Frequec Soluto: f f Total

64 7 Eample Fd the momet coeffcet of skewess ad kurtoss for the followg frequec dstrbuto. Class Frequec m = s = 8.575, m 3 = -.693, m 4 =

65 Chapter Seve Itroducto to Probablt Theor Termolog ad Notatos Elemetar Set Theor Epermet: A radom epermet s a operato whose output caot be predcted wth certat. It s the process that geerates observatos. Eample: Tossg a co oce or several tmes, selectg a card or cards from a deck, obtag blood tpes from a group of dvduals etc. Outcome: Output of a epermet s called outcome. The umber of outcomes depeds upo the ature of the epermet. Sample Space: The set of all possble outcomes of a epermet s called the sample space of the epermet, ad s deoted b Ω. Each dstct outcome s called elemet or pot of the sample space. A sample space s sad to be dscrete f t cossts of a fte umber of sample pots or coutabl fte sample pots. A set s called coutable f ts elemets ca be placed a oe-to-oe correspodece wth the postve tegers. Eample. I a epermet of tossg a co, the possble outcomes are Ω = {H, T}.If we have two cos, the we would get Ω = {HH, HT, TH, TT}. Eample. I the epermet of rollg a de, the sample space s Ω = {,, 3, 4, 5, 6}. If the epermet cossts of tossg two dce, the the sample space cossts of the 36 pots. Ω = {(, j):, j =,, 3, 4, 5, 6}.

66 Eample 3. I the epermet of tossg a co repeatedl ad coutg the umber of tosses utl the frst head appears. Ω = {,, 3 } Note that there are a fte umber of outcomes. Eample 4. I the epermet cossts of measurg the lfetme t ( hours) of a electroc equpmet Ω = {t: 0 t < } whch s cotuous ad fte. Evet: A evet s a collecto (subset) of outcomes cotaed the sample space Ω,.e. A s a evet ff. A evet s sad to be smple f t cossts of eactl oe outcome ad compoud f t cossts of more tha oe outcome. Eample: Whe a sgle regular de s rolled oce Ω = {,, 3, 4, 5, 6}. The each subset A = {}, B = {, 4, 6}, C = {, 3, 5}, D = {,, 4, 5}. A s a smple evet whle B, C ad D are compoud evets. Some Relatos from Set Theor: Let ow Ω be a abstract uversal set ad A, B etc. deote sets, collectos of elemets Ω.. meas that a elemet ω belogs to a set A. meas that ω does ot belog to a set A.. deotes the empt set, whch has o elemets. 3. A c s the complemet set of A. It cossts of all elemets ω that do ot belog to A. 4. A B meas that A s a subset of B. Ths meas that f, the. I addto, we have for a set A Ω. Note that A B ad B A f ad ol f A = B. We use also o occaso the otato of strct cluso, A s

67 3 a proper subset of B deoted b A B, whch meas that all elemets of A are B, but ot all elemets of B are A ( A B). 5. If A B, the B c A c. 6. The uo of two sets A ad B deoted b A B s the evet cosstg of all elemets ω such that or or both evets. For a sequece of sets A, A, the uo cossts of elemets ω such that there s at least oe A such that. 7. The tersecto of two sets A ad B deoted A B (or AB) s the evet cosstg of all elemets ω such that ad. For a sequece of sets A, A, the tersecto cossts of the elemets ω such that for all. Mutuall Eclusve Evets: (Dsjot evets) Two evets A ad B are sad to be mutuall eclusve or dsjot f the occurrece of A precludes the occurrece of B ad vce-versa, A B = Ø.e., the caot occur smultaeousl. The sets A, A,, A are parwse dsjot f all pars A, A j are dsjot for j. Eample: - I the radom epermet of tossg a co, the two evets A = {H}, B = {T} are mutuall eclusve. - Suppose that Ω = {,, 3, 4}. If: A = {, }, B = {, 3, 4} ad C = {4} The, A ad B are subsets of Ω; A Ω ad B Ω, A, 3A, A B = {,, 3, 4}, A B = {}, A C = Ø, A c = {3, 4}, A A c = Ω, A ad C are dsjot or mutuall eclusve.

68 4 Ve dagrams: A Ve dagram s a graphcal represetato that s ofte useful for dscussg the cocepts of set relatoshps: A B A B A c A B = Ø Laws of the Set Relatos:. Commutatve Laws A B = B A A B = B A. Assocatve Laws A (B C) = (A B) C A (B C) = (A B) C 3. Dstrbutve laws A (B C) = (A B) (A C) A (B C) = (A B) (A C) 4. Idett Laws A Ø = A A Ø = Ø A Ω = Ω A Ω= A 5. Complemet laws A A c = Ω A A c = Ø (A c ) c = A Ω c = Ø Ø c = Ω 6. Demorga`s Laws (A B) c = A c B c ad (A B) c = A c B c (A B) c (A B) c

69 5 For a sequece of sets A, A, ( ) ad ( ) Power set of Ω: The faml of all subsets of Ω s kow as the power set of Ω. (Ω) = {A: A Ω} I geeral, f Ω s a fte set ad has elemets the (Ω) wll have elemets. Eample: Ω = {a, b} the (Ω) = {{a}, {b}, Ω, ϕ} Algebras of sets: Evet space: A evet space s a sgma feld obtaed from a sample space. A elemet of the evet space s called a evet. A evet probablt correspods to a set set theor. Remark:. The sample space Ω s called the sure evet.. The empt set ϕ s called the mpossble evet.. The operatos,,, o a umber of evets are also evets. Sgma Feld: Let Ω deote a uversal set. A collecto of Ω s called a σ- algebra or a σ- feld f. ( ). If A, the A c 3. If A, A,,A,, the of subsets Eamples:. = {ϕ, Ω} s the smallest σ feld (smallest evet space).. ( ) s the largest σ feld o Ω.

70 6 3. Let A Ω, the = {A, A c, Ω, ϕ} s the σ feld geerated b the set A. 4. Let Ω = {,, 3}. The = {{}, {, 3}, Ω, ϕ} s a σ feld. But {{}, {}, {3},{,, 3}, ϕ} s ot a σ feld. Propertes of σ- Felds Let A be a σ feld over Ω the:. Proof: Sce s a σ feld, there est a set A Hece A c. Cosder the fte sequece of elemets A, A c, A c, A c, but. Hece. Proof: Sce But Ω c = ϕ, hece the 3. ( ) Proof: Sce for all, the for all. Ad So ( ) Ad b De Morga s law 4. ( ) Proof: Cosder the fte sequece of elemets A, A, A 3,, A, ϕ, ϕ, ϕ, whch belog to The

71 7 Hece 5. ( ) Proof: Sce the ad ( ) Ad b De Morga s law as requred. Measurable Space: The par (Ω, ) of a sample space Ω ad a evet space s called a measurable space. Sgma feld geerated b Ȼ Gve a sample space Ω, cosder a arbtrar class Ȼ of subsets of Ω. The smallest sgma feld cotag all of the sets Ȼ s called the sgma feld geerated b Ȼ ad s deoted b σ(ȼ). Here, b smallest we mea that f a sgma feld cotas Ȼ, the t also cotas σ (Ȼ). s geerated b the class Ȼ over Ω meas that. s a σ feld over Ω.. 3. If s a σ feld over Ω, such that Ȼ the. Eample: Suppose that Ω = {a, b, c}. Let Ȼ = {{b}}, Ȼ = {{a}, {b}}, Ȼ 3 = {{a}, {c}} Fd (Ω), σ-(ȼ ), σ-(ȼ ) σ-(ȼ 3 ) (Ω)= {{a}, {b}, {c}, {a, b}, {a, c}, {b, c}, Ω, ϕ} σ-(ȼ )={{b}, {a, c}, Ω, ϕ} σ-(ȼ )= {{a}, {b}, {b, c}, {a, c},{a, b}, {c}, Ω, ϕ} σ-(ȼ 3 )= {{a}, {c}, {b, c}, {a, b}, {a, c}, {b}, Ω, ϕ} H.W.. If A = {,, 3} fd the power set of A.. Let Ω = {a, b, c, d} ad let Ȼ = {{a}, {c, d}}, fd σ-(ȼ).

72 8 Probablt measure space: A probablt space s gve b the trple (Ω,, P) where Ω s a set of outcomes, s set of subsets of Ω, the set of possble evets ad P s a fucto assgg probabltes to evets P:. s take to be a σ- feld. Probablt measure: For the measurable space (Ω, ), a set fucto s called a probablt measure f t assgs a real umber P(A) to a set P: uder the costrat of the followg aoms. If, P(A) 0. P(Ω)= 3. ( ) ad A A j = ϕ j the ( ) ( ) Theorem: Let (Ω,, P) be a probablt space, the. P(ϕ) = 0 Proof: Cosder the fte sequece of evets A, A, such that A = ϕ for =,, ; the b aom 3 ( ) ( ) ( ) ( ), whch ca hold ol f P(ϕ) = 0. For a fte sequece of dsjot evets A, A,, A ( ) ( ) Proof: Cosder the fte sequece of evets A, A, whch A, A,, A are the gve dsjot evets ad A = ϕ for >. The

73 9, ad b aom 3 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3. For ever evet A, P(A c ) = P(A) A A c = Ω ad A A c = ϕ P(Ω) = P(A A c ) = P(A) + P(A c ) Sce, P(Ω) = Hece, P(A c ) = P(A). 4. If A ad B are evets, ad A B, the P(A) P(B) Proof: Sce A B, we ca wrte B = A (A c B) P(B) = P(A) + P(A c B) Sce P(A c B) 0 Hece, P(B) P(A) A c B ( A ad A c B are mutuall eclusve) 5. For a evet A, 0 P(A) for a evet A. Proof: For a evet A Ω we have Ø A Ω P(Ø) P(A) P(Ω)

74 0 Sce P(Ø) = 0 ad P(Ω) = Hece, 0 P(A). 6. For a two evets A ad B, P(A B) = P(A) + P(B) - P(A B). Proof: A B = A (B A c ) P(A B) = P(A) + PB A c ) B = (B A) (B A c ) (Dsjot) P(B) = P(B A) + P(B A c ) (Dsjot) B A c P(B A c ) = P(B) - P(B A) P(A B) = P(A) + P(B) - P(B A) Ad f A ad B are mutuall eclusve, the P(A B) = P(A) + P(B), sce P(A B) = 0 7. If A, B, ad C are a three evets, the P(A B C) = P(A)+P(B)+P(C) P(A B) P(A C) P(B C)+ P(A B C) Proof: A B C = A (B C) P(A B C) = P(A) + P(B C) P[A (B C)] = P(A) + [P(B) + P(C) P(B C)] P[(A B) (A C)] = P(A) + P(B) + P(C) P(B C) [P(A B) + P(A C) P(A B C)] = P(A) + P(B) + P(C) P(A B) P(A C) - P(B C) + P(A B C)

75 Cotut from below ad cotut from above: A probablt measure has further the followg propertes: Let A, =,, be a fte sequece of evets. If A A, the ( ) ( ). If, the ( ) ( ) Proof: () where = ϕ ( ) ( ) ( ) ( ) ( ) ( ) () ( ) ( ) ( ) ( ) ( ) ( ) () From () ad () we get; ( ) ( ) Proof: () = ( ) ( ) B De Morga s law ( ) ( ) ( ) ( ) Therefore ( ) ( ( )) ( )

76 ( ) ( ) Hece, ( ) ( ) Equprobable Space: P s a equprobable fucto o (Ω,. Ω s a o empt fte set.. P: ( ), such that ( ) ( ) Problem: Verf that P s a probablt measure fucto. P: ( ) ( )), f. ( ), ( ). ( ) 3. Let be fte sequece of dsjot evets (A A j = ϕ j) belog to ( ). Sce Ω s fte A +k = ϕ the ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Thus (Ω, ( ), P) s a probablt measure space.

77 3 Relatve frequec defto of probablt (Statstcal probablt): If represets suffcetl large umber of trals made to see whether a evet A occurs or ot, ad m represets the umber of trals whch A s observed to occur the the probablt of m occurrece of A s gve b P(A) =.Ths s called statstcal lm or emprcal probablt. Clearl 0 P(A) where P(A) = 0 sgfes that the evet s mpossble whle P(A) = sgfes that t s certa. Note: The fact that P(A) = 0 does ot mpl that A = Ø. Eample: A epermet cossts of throwg two ordar dce, oe de s red ad the other s clear. Ω = {(, ) =,,, 6; =,,, 6 }. a. What s the probablt of throwg a double? E = {(, ), (, ),, (6, 6)} 6 p(e ) = = 36 6 b. What s the probablt that the umber o the clear de s at least 3 greater tha the umber o the red de? E = {(, 4), (, 5), (, 6), (, 5), (, 6), (3, 6)} 6 p(e ) = = = 36 6 c. What s the probablt that the sum of both faces s0? E 3 = {(4, 6), (6, 4), (5, 5)} 3 p(e 3 ) = = = 36 d. What s the probablt that r 3 or c? Let A cossts of 8 pots of Ω for whch r 3. B cossts of pots of Ω for whch c.

78 4 A = {(3, ), (3, ),, (3, 6),(, ), (, ),, (, 6), (, ), (, ),, (, 6)}. B = {(, ), (, ),, (6, ), (, ), (, ),, (6, )}. A B = {(3, ), (3, ), (, ), (, ), (, ), (, )} p(a or B) = p(a B) = p(a) + p(b) - p(a B) = e. What s the probablt that the sum r + c s 7 or 0? Let A cossts of the umber of pots of Ω for whch the sum s 7. B cossts of the umber of pots of Ω for whch the sum s 0. A = {(, 6), (6, ), (, 5), (5, ), 3, 4), (4, 3)}, B = {(4, 6), (6, 4), (5, 5)} A B = Ø (mutuall eclusve) p(a B) = p(a) + p(b) = Eample: Let tems be chose at radom from a lot cotag tems of whch 4 are defectve. Let A = {both tems are defectve} B = {both tems are o-defectve} C = {At least oe tem s defectve} E = {At least oe tem s o-defectve Fd: p(a), p(b), p(c) ad p(e). 4 Ω ca occur C = 66 was whch tems ca be chose from.the umber of was that defectve tems be chose from 4 defectve tems s 4 C = 6

79 5 The umber of was that o-defectve tems be chose from 8 odefectve tems s p(a) = C C C = 8. 8 C 8 4 p(b) = C Sce C s the complemet of B, p(c) = p(b c 8 ) = p(b) = 66 Sce E s the complemet of A, p(e) = p(a c ) = p(a) = Eample: A ur cotas 0 black, 5 whte, ad 5 red balls. What s the probablt of drawg a black, a whte or a red ball? Let B = {the ball draw s black}, W = {the ball draw s whte}, R = {the ball draw s red} p(b) = p(w) = p(r) = Eample: A card s draw from a deck. a. What s the probablt that the card draw s a heart? b. What s the probablt that the card draw s a ace? Let A = {the card draw s a heart}, B = {the card draw s a ace} 3 4 p(a) = p(b) =

80 6 Eample: A class cotas 0 me ad 0 wome of whch half the me ad half the wome have brow ees. Fd the probablt that a perso chose at radom s a ma or has brow ees. 0 Let A = {perso s a ma} p(a) = B = {perso has brow ees} p(b) = 30 A B = {perso s a ma ad has brow ees} p(a B) = p(a B) = p(a) + p(b) - p(a B) 4 = H.W.. A far de s rolled twce, fd the probablt that: a. The sum of both outcomes s 7. b. The sum s ot 7. c. The sum s more tha 7. d. The sum s less tha or equal to 7.. A card s draw from a deck. What s the probablt that: a. The card s red. b. The card s a damod. c. The card s a pcture. d. The card s a pcture or damod. e. The card s a joker.

81 7 3. Cosder two evets A ad B such that P(A) = /3, ad P(B) = /. Determe P(B A c ) for each of the followg cases: a. A ad B are dsjot. b. A s a subset of B. c. P(AB) = /8 4. For a evet A, f P(B) =, show that P(A) = P(AB) Ht: Take A = AB AB c 5. Cosder the epermet whch a far co s tossed oce ad a balaced de s rolled oce. a. Descrbe the sample space. b. What s the probablt that a head s appeared o the co, ad a odd umber s appeared o the de? 6. A school cotas studets grades,, 3, 4, 5, ad 6. Grades, 3, 4, 5, ad 6 all cota the same umber of studets, but there s twce ths umber grade. A studet s selected at radom from a lst of all the studets the school. a. What s the probablt that the selected studet s grade 3? b. What s the probablt that the selected studet s a odd umbered grade?

82 8 Codtoal Probablt: Let A ad B be two evets of the probablt space (Ω,, P), the codtoal probablt of evet A gve that evet B has occurred s defed b: P( A B) P( AB) P(A B) = or P( B) P( B) f P(B) > 0 Problem: For gve evet B for whch P(B) > 0, show that P( B) s a probablt measure fucto. Soluto: P( AB). P(A B) = P( B) 0 for ever A. P( B) P( B). P(Ω B) = P( B) P( B) 3. If A, A, s a sequece of mutuall eclusve evets ad, the ( ) ( ) (( ) ) ( ) ( ( )) ( ) Hece, P( B) s a probablt measure fucto. ( ) ( ) Eample : Three far cos are tossed. Fd the probablt that the are all heads f:. The frst co s head. Let A = {the frst co s head} = {HHH, HHT, HTH, HTT} B = {the three cos are heads} = {HHH}, A B = {HHH} p( AB) # AB 8 P(B A) = p( A) # A 4 4 8

83 9. At least oe of the cos s heads. Let A= {HTT, THT, TTH, HHH, HHT, HTH, THH} B = {HHH}, A B = {HHH} P(B A) = p( AB) p( A) Eample : A far de s throw, f the umber appearg s less tha 4. What s the probablt that:. The umber s odd.. The umber s. Let B = {the umber s less tha 4} = {,, 3} A = {the umber s odd} = {, 3, 5}, A B = {, 3} p( AB) 6 P(A B) = p( B) Let C = {the umber s } = {}, B C = {} p(c B) = p( CB) p( B) Eample 3: Two dgts are selected at radom from,, 9.If the sum s eve, fd the probablt that both umbers are odd. 9 There are C = 36 was to choose umbers from {,,, 9}. The sum s eve f both umbers are eve or both are odd. There are 4 eve umbers {, 4, 6, 8}. Hece there are to choose two eve umbers. There are 5 odd umbers {, 3, 5, 7, 9}. Hece there are was to choose two odd umbers. 4 C = 6 was 5 C = 0

84 0 Thus, there are = 6 was to choose two umbers such that the sum s eve. Let A = {the sum of the two dgts s eve} 6 elemets 6 P(A) = 36 B = {the two dgts are odd} 0 elemets, 0 P(B) = 36 A B = B 0 elemets 0 P( AB) P(B A) = P( A) Propertes of P( B) : Assume that the probablt space (Ω, satsf P(B) > 0.. If A = B, the P(A B)=, P) s gve, ad let B. P(ϕ B) = 0 ad P(B c B) = 0 3. If A, A,, A are mutuall eclusve evets, the ( ) ( ) 4. For a two evets A ad A, ( ) ( ) ( ) ( ) 5. P(A B) + P(A c B) = 6. If A ad A, ad, the P(A B) P(A B)