Chapter 1 Data and Statistics

Transcription

1 Chapter Data ad Statstcs Motvato: the followg kds of statemets ewspaper ad magaze appear very frequetly, Sales of ew homes are accrug at a rate of 7000 homes per year. The uemploymet rate has dropped to 4.0%. The Dow Joes Idustral Average closed at 0000 Cesus The above umercal descrptos are very famlar to most of us sce we use t everyday lfe. As a matter of fact, these are part of statstcs. Therefore, statstcs s our everyday lfe. We ow gve a descrpto about statstcs. Defto of statstcs: Statstcs s the art ad scece of collectg, aalyzg, presetg, terpretg ad predctg data. Objectve of ths course: usg statstcs s to gve the maagers ad decso makers a better uderstadg of the busess ad ecoomc evromet ad thus eable them to make more formed ad better decso.. Data I Bass compoets of a data set: Usually, a data set cossts the followg compoets: Elemet: the ettes o whch data are collected. Varable: a characterstc of terest for the elemet. Observato: the set of measuremets collected for a partcular elemet. Eample : We have a data set for the followg 5 stocks: Stock Aual Sales mllo Eargs per share $ Echage where to trade Cache Ic OTC Koss Corp OTC

2 ar Techology NYSE Scetfc Tech OTC Wester Beef OTC Note: OTC stads for over the couter whle NYSE stads for New York Stock Echage. I the above data set, Elemets Varables Observatos Cache Ic., Koss Corp, ar Techology, Scetfc Tech, Wester Beef Aual Sales, Eargs per share, Echage 86.6,0.5,OTC,6.,0.89,OTC,8.,0.,NYSE, 7.,0.46,OTC,7.7,0.78,OTC II Qualtatve ad Quattatve Data: Qualtatve data: labels or ames used to detfy a attrbute of each elemet. Quattatve data: dcatg how much or how may Eample cotue: Qualtatve data: OTC, OTC, NYSE, OTC ad OTC Quattatve data: 86.6, 6., 8., 7., 7.7, 0.5, 0.89, 0., 0.46 ad 0.78 The varable Echage s referred to as a qualtatve varable. The varables Aual Sales ad Eargs per share are referred to as quattatve varables. Note: quattatve data are always umerc, but qualtatve data may be ether umerc or oumerc, for eample, d umbers ad automoble lcese plate umbers are qualtatve data. Note: ordary arthmetc operatos are meagful oly wth quattatve data ad are ot meagful wth qualtatve data. III Cross-Sectoal ad Tme Seres Data: Cross-sectoal data: data collected at the same or appromately the same pot tme. Tme seres data: data collected over several tme perods.

3 Ole Eercse: Eercse.. Eercse... Data Source: There are two sources for data collecto, oe s estg sources ad the other s statstcal studes. I Estg Sources: There are two estg sources: Compay: some of the data commoly avalable from the teral formato sources of most compay cludg employee records, producto records, vetory records, sale records, credt records ad customer profle, etc Govermet Agecy: Departmet of Labor, Bureau of the Cesus, Federal Reserve Board, Offce of Maagemet ad Budget, Departmet of Commerce. II Statstcal Studes: A statstcal study ca be coducted to obta the data ot readly avalable from estg sources. Such statstcal studes ca be classfed as ether epermetal or observatoal. Epermetal Study: attempt s made to cotrol or fluece the varables of terest, for eample, drug test ad dustral product test. Observatoal Study: o attempt s made to cotrol or fluece the varable of terest, for eample, survey. Ole Eercse: Eercse... Descrptve Statstcs: There are two classes of descrptve statstcs, oe class cludes table ad graph ad the other class cludes umercal measures ad de umbers. I Tabular ad Graphcal Approaches:

4 Eample cotue: Tabular approach for stock data: Echage Frequecy ercet OTC 4 80% NYSE 0% 5 00% Graphcal approach for stock data: OTC NYSE II Numercal Measures ad Ide Numbers: Some umercal quattes ca be used to provde mportat formato about the data, for eample, the average or mea. Ide umbers are wdely used busess, for eample, the Cosumer rce Ide CI ad te Dow Joes Idustral Average DJIA. Ole Eercse: Eercse.. Eercse...4 Statstcal Iferece: Descrptve statstcs troduced secto.4 ca provde mportat ad tutve formato about the data of terest. However, these statstcal measures are maly 4

5 eploratory. For more detaled, rgorous ad accurate results, the statstcal ferece procedure s requred. To coduct a statstcal ferece, data eed to be draw from a set of elemets of terest. We ow troduce some basc compoets the statstcal ferece procedure. They are: opulato: the set of all elemets of terest a partcular study. Sample: a subset of the populato. Data from a sample ca be used to make estmates ad test hypotheses about the characterstcs of a populato Eample : Suppose there are bulbs produced a bulbs factory. Objectve: wat to kow the average lfetme of the bulbs. The bulbs are the populato of terest. I practce, t s ot possble also ot realstc to test bulbs for the lfetme. Oe workable way s to draw a sample, say 00 bulbs, ad the test for ther lfetme. Suppose the average lfetme of the 00 bulbs s 750 hours. The, the estmate guess of the average lfetme of the bulbs s 750 hours. Note: the process of makg estmates ad testg hypotheses about the characterstcs of a populato s referred to as statstcal ferece. Ole Eercse: Eercse.4. Chapter Descrptve Statstcs: Table ad Graph The logcal flow of ths chapter: Summarzg qualtatve data usg tables ad graphs. Summarzg quattatve data usg tables ad graphs. 5

6 Eploratory data aalyss usg smple arthmetc ad easy-to-draw graphs such that the data ca be summarzed quckly.. Summarzg Qualtatve Data: For qualtatve data, we ca use frequecy dstrbuto ad relatve frequecy. We ow troduce frequecy dstrbuto, relatve frequecy ad percet frequecy. Frequecy dstrbuto: tabular summary of data dcatg the umber of data values each of several ooverlappg classes. Relatve frequecy: frequecy of a class/, where s the total umber of the data. ercet frequecy: relatve frequecy 00%. Based o the frequecy dstrbuto, relatve frequecy, ad percet frequecy of the data, we ca use table ad graphs to dsplay these frequeces. Eample: Forbes vestgates the degrees of 5 best pad CEO chef eecutve offcer. Tabular summary: Degrees Frequecy Relatve Frequecy ercet Frequecy Noe Bachelor Master Doctorate Total

7 Graphcal dsplay: Bar Graph: Noe Bachelor Master Doctorate e Graph: CEO Degrees Noe Bachelor Master Doctorate Note: most statstcas recommed that from 5 to 0 classes be used a frequecy dstrbuto; classes wth smaller frequeces should ormally be grouped!! 7

8 Ole Eercse: Eercse.. Eercse... Summarzg Quattatve Data: Determe the classes: For quattatve data, we eed to defe the classes frst. There are steps to defe the classes for a frequecy dstrbuto: Step : Determe the umber of ooverlappg classes, usually 5 to 0 classes. Step : Determe the wdth of each class, class wdth largest data value smallest data umber of classes value Note: the umber of classes ad the appromate class are determed by tral ad error!! Step : Determe the class lmts: the smallest possble data value should be larger tha the lower class lmt whle the largest possble data value should be smaller tha the upper class lmt. Eample: Suppose we have the followg data days: We appled the above procedure to ths data. Step : We choose 5 to be the umber of classes. 8

9 Step : largest data value smallest data value class wdth 4. umber of classes 5 Therefore, we use 5 as the class wdth. Step : The 5 classes we choose are Note: the lower class lmt the frst class 0 s smaller tha the smallest data value. Also, the upper class lmt the last class 4 s smaller tha the largest data value. Summarzg quattatve data: Tabular summary: I addto to frequecy, relatve frequecy ad percet frequecy, aother tabular summary of quattatve data s the cumulatve frequecy dstrbuto. Cumulatve frequecy dstrbuto: the umber of data tems wth values less tha or equal to the upper class lmt of each class. Graphcal dsplay: I addto to hstogram, aother graphcal dsplay of quattatve data s ogve. Ogve: the umber of data tems wth values less tha or equal to the upper class lmt of each class. Eample cotue: Classes Frequecy Relatve Frequecy ercet Frequecy

10 Total 0 00 Classes Cumulatve Cumulatve Relatve Cumulatve ercet Frequecy Frequecy Frequecy The hstogram s data The ogve plot s 0

11 Ogve plot 0 cumulatve frequecy data Ole Eercse: Eercse.. Eercse... Eploratory Data Aalyss: Stem-ad-leaf dsplay s a useful eploratory data aalyss tool whch ca provde a dea of the shape of the dstrbuto of a set of quattatve data. Eample: Suppose the followg data are the mdterm scores of 0 studets, 7,, 9, 8, 95, 87, 66, 68, 7, 5. The, the stem-ad-leaf dsplay s

12 Ole Eercse: Eercse.. Chapter Descrptve Statstcs: Numercal Methods Suppose y, y,, y K N are all the elemets the populato ad,, K, are the sample draw from y, y,, y K N, where N s referred to as the populato sze ad s the sample sze. I ths chapter, we troduce several umercal measures to obta mportat formato about the populato. These umercal measures computed from a sample are called sample statstcs whle those umercal measures computed from a populato are called populato parameters. I practce, t s ot realstc or ot possble to obta populato parameter from a populato, for eample, the average lfetme of bulbs. Therefore, the sample statstc ca be used to estmate the populato parameter, for eample, the average lfetme of 00 bulbs ca be used to estmate the average lfetme of bulbs... Measure of Locato: Eample: Suppose the followg data are the scores of 0 studets a quz,,, 5, 7, 9,, 4, 6, 8, 0. Some measures eed to be used to provde formato about the performace of the 0 studets ths quz. I Mea:

13 Sample mea: sample statstc opulato mea: parameter N N y µ populato Bascally, the mea ca provde the formato about the ceter of the data. Itutvely, t ca measure the rough locato of the data. Eample cotue: + + L II Meda: The data are arraged ascedg or descedg order. The,. As the sample sze s odd, the meda s the mddle value.. As the sample sze s eve, the meda s the mea of the mddle two umbers. Eample cotue: meda If the data are,,, 4, 5, 6, 7, 8, 9, 0,. The, meda Note: the meda s less sestve to the data wth etreme values tha the mea. For eample, the prevous data, suppose the last data has

14 bee wrogly typed, the data become,, 5, 7, 9,, 4, 6, 8, 00. The the meda s stll 5.5 whle the mea becomes 4.5. III Mode: The data value occurs wth greatest frequecy ot ecessarly to be umercal. Note: f the data have eactly two modes, we say that the data are bmodal. If the data have more tha two modes, we say that the data are multmodal. IV ercetle: The pth percetle s a value such as at least p percet of the data have ths value or less. Note: 50th percetle meda!! The procedure to calculate the pth percetle:. Arrage the data ascedg order. p. Compute a de,. 00. a If s ot a teger, roud up,.e., the et teger value greater tha deote the posto of the pth percetle. b If s a teger, the pth percetle s the average of the data values postos ad +. Eample cotue: lease fd 40th percetle ad 6th percetle for the prevous data. [Soluto] Step : the data ascedg order are,,, 4, 5, 6, 7, 8, 9, 0. 4

15 Step : For 40 th percetle, For 6 th percetle, Step : ad th percetle th percetle V Quartles: Whe dvdg data to 4 parts, the dvso pots are referred to as the quartle!! That s, Q the frst quartle or 5th percetle Q the secod quartle or 50th percetle Q the thrd quartle or 75th percetle Eample cotuous: Fd the frst quartle ad the thrd quartle for the prevous eample. Step : For the frst quartle, For the thrd quartle, Step : Q

16 ad Q 8 Ole Eercse: Eercse.. Eercse... Measure of Dsperso: Eample: Suppose there are two factores producg the batteres. From each factory, 0 batteres are draw to test for the lfetme hours. These lfetmes are: Factory : 0., 9.9, 0., 9.9, 9.9, 0., 9.9, 0., 9.9, 0. Factory : 6, 5, 7, 4, 6, 5,,, 9,. The mea lfetmes of the two factores are both 0. However, by lookg at the data, t s obvous that the batteres produced by factory are much more relable tha the oes by factory. Ths mples other measures for measurg the dsperso or varato of the data are requred. I Rage: ragelargest value of the data-smallest value of the data. Eample cotue: Rage of lfetme data for factory Rage of lfetme data for factory 6- The rage of battery lfetmes for factory s much smaller tha the oe for factor. Note: the rage s seldom used as the oly measure of dsperso. The 6

17 rage s hghly flueced by a etremely large or a etremely small data value. II Iterquartle Rage: Iterquartle s the dfferece betwee the thrd ad the frst quartles. That s, IQR. Q Q Eample: The frst quartle ad the thrd quartle for the data from factory are 9.9 ad 0., respectvely, ad 6 ad 4 for the data from factory. Therefore, IQR factory IQR factory The terquartle of battery lfetmes for factory s much smaller tha the oe for factor. III Varace ad Stadard Devato: populato devato about the mea: y µ,,, K, N sample devato about the mea:,,, K, Itutvely, the populato devato ad the sample devato ca measure how far the data s from the ceter of the data. The, populato varace ad sample varace are the sum of square of the populato devato ad sample devato, ad σ N y µ N s, 7

18 respectvely. The populato stadard devato ad sample stadard devato are the square root of populato varace ad sample varace: ad σ σ s s, respectvely. Large sample varace or sample stadard devato mples the data are dspersed or are hghly vared. 0 Note: Eample: s s factory. factory L L The sample varace of battery lfetmes for factory s 90 tmes larger tha the oe for factor. The sample stadard devato for the data from factores ad are respectvely ad , IV Coeffcet of Varato: The coeffcet of varato s aother useful statstc for measurg the dsperso of the data. The coeffcet of varato s 8

19 C s. V. 00 The coeffcet of varato s varat wth respect to the scale of the data. O the other had, the stadard devato s ot scale-varat. The followg eample demostrates the property. Eample: I the battery data from factory, suppose the measuremet s mutes rather tha hours. The, the data are 606, 594, 606, 594, 594, 606, 594, 606, 594, 606. Thus, the stadard devato becomes 6.45 whch s 60 tmes larger tha the oe based o the orgal data measured hours. However, o matter the data are measured hours ad mutes, the coeffcet of varato s C. V Note: sce the coeffcet of varato s scale-varat, t s very useful for comparg the dsperso of dfferet data. For eample, the prevous battery data, f the lfetme of the batteres from factory ad factory are measured mutes ad hours, respectvely, the stadard devato for factory, 6.45, would be larger tha for factory, However, the coeffcet of varato for factory,.054 s stll much smaller tha the oe for factory, Ole Eercse: Eercse.. Eercse... Eploratory Data Aalyss: I Fve-Number Summary: The fve umber summary ca provde mportat formato about both the locato 9

20 ad the dsperso of the data. They are Smallest value Frst quartle Meda Thrd quartle Largest value Eample cotue: The orgal data hours are: Factory : 0., 9.9, 0., 9.9, 9.9, 0., 9.9, 0., 9.9, 0. Factory : 6, 5, 7, 4, 6, 5,,, 9,. The fve-umber summary for the data from both factores s Smallest Q Meda Q Largest Factory Factory II Bo lot: The bo-plot s commoly used graphcal method to provde formato about both the locato ad dsperso of the data. Especally, as the terest s the comparso of the data from dfferet populatos, the bo-plot ca provde sght. The bo-plot s.5iqr.5iqr lower IQR upper lmt Q Q lmt Note: data outsde upper lmt ad lower lmt are called outlers. Eample cotue: The bo-plot for the data from the two factores s 0

21 factory factory Ole Eercse: Eercse...4 Measures of Relatve Locato: z-score s the quatty whch ca be used to measure the relatve locato of the data. Z-score, referred to as the stadardzed value for observato, s defed as z s. Note: z s the umber of stadard devato from the mea. Eample cotue: Factory :

22 z Factory : z There are two results related to the locato of the data. The frst result s Chebyshev s theorem. Chebyshev s Theorem: For ay populato, wth k stadard devato of mea, there are at least k 00 % of the data, where k s ay value greater tha. Based o Chebyshev s theorem, for ay data set, t could be roughly estmated that at least 00% of data wth k sample stadard devato of mea. k Eample cotue: As k, based o Chebyshev s theorem, at least 00 % 75 % of the data are estmated wth stadard devatos of mea. For the data from factory ad factory, all the data are wth sample devatos of mea,.e., all the data have z-score wth absolute values smaller tha. The secod result s based o the emprcal rule. The rule s especally applcable as the data have a bell-shaped dstrbuto. The emprcal rule s Appromately 68% of the data wll be wth oe stadard devato of the mea z. Appromately 95% of the data wll be wth oe stadard devato of the

23 mea z. Almost all of the data wll be wth oe stadard devato of the mea z. Eample cotue: For data from factory, all the data are wth oe stadard devato of the mea whle 60% of the data are wth oe stadard devato of the mea for the data from the factory. The result based o the emprcal rule s ot applcable to the two data set sce the two data sets are ot bell-shaped. However, for the followg data, The hstogram of the above data gve below dcates the data s roughly bell-shaped r Appromately 65% of the data are wth oe stadard devato of the mea, whch s smlar to the result based o the emprcal rule 68%. Detectg Outlers: To detfy the outlers, we ca use ether the bo-plot or the z-score. The outlers detfed by the bo-plot are those data outsde the upper lmt or lower lmt whle the outlers detfed by z-score are those wth z-score smaller tha or greater tha.

24 Note: the outlers detfed by bo-plot mght be dfferet from those detfed by usg z-score. Ole Eercse: Eercse.4..5 The Weghted Mea ad Grouped Data: Weghted Mea: w w w. Note: whe data values vary mportace, the aalyst must choose the weght that best reflects the mportace of each data value the determato of the mea. Eample : The followg are 5 purchases of a raw materal over the past moths. urchase Cost per oud $ Number of ouds Fd the mea cost per poud. [solutos:] 4

25 w 00, w 500, w 750, w4 000, w ad.00,.40,.80, 4.90, 5.5. The, w w w opulato Mea for Grouped Data: where µ g m F F M k k k k m k k m F N k M k, M : the mdpot for class k, k m F : the frequecy for class k the populato, k N F k : the populato sze. k Sample Mea for Grouped Data: 5

26 where g m f M k k k k m k f k m f k M k, f : the frequecy for class k the sample, k m f k : the sample sze. k opulato Varace for Grouped Data: σ g m k F k M µ N k g Sample Varace for Grouped Data: s g m k f k M k g m k f k M k g Eample : The followg are the frequecy dstrbuto of the tme days requred to complete year-ed audts: Audt Tme days Frequecy

27 What s the mea ad the varace of the audt tme? [solutos:] f 4, f 8, f 5, f4, f5. f + f + f + f4 + f ad M, M 7, M, M 4 7, M 5. Thus, ad g f M f s g f M g Ole Eercse: Eercse.5. Chapter 4 Assocato Betwee Two Varables I ths chapter, we troduce several methods to measure the assocato. They are: Crosstabulatos ad scatter dagrams 7

28 Numercal measures of assocato 4. Crosstabulatos ad Scatter Dagrams: The crosstabulato table ad the scatter dagram graph ca help us uderstad the relatoshp betwee two varables.. Crosstabulatos Eample: Objectve: eplore the assocato of the qualty ad the prce for the restaurats the Los Ageles area. The followg table s the crosstabulato of the qualty ratg good, very good ad ecellet ad the mea prce $0-9, $0-9, $0-9, ad $40-49 data collected for a sample 00 restaurats located the Los Ageles area. Qualty Ratg Meal rce $0-9 $0-9 $0-9 $40-49 Total Good Very Good Ecellet Total The above crosstabulato provdes sght abut the relatoshp betwee the varables, qualty ratg ad mea prce. It seems hgher meal prces appear to be assocated wth the hgher qualty restaurats ad the lower meal prces appear to be assocated wth the lower qualty restaurats. For eample, for the most epesve restaurats $40-49, oe of these restaurats s rated the lowest qualty but most of them are rated hghest qualty. O the other had, for the least epesve restaurats $0-9, oly of these restaurats are rated the hghest qualty.56% but over half 78 8

29 of them are rated lowest qualty.. Scatter Dagram Suppose we have the followg scatter dagrams for the weghts ad heghts of the studets: Scatter Dagram of Weght v.s. Heght heght heght heght weght weght weght The left scatter dagram dcates the postve relatoshp betwee weght ad heght whle the rght scatter dagram mples the egatve relatoshp betwee the two varables. The mddle scatter dagram shows that there s o apparet relatoshp betwee the weght ad heght. Ole Eercse: Eercse Numercal Measures of Assocato: There are several umercal measures of assocato. We frst troduce the covarace of two varables. I Covarace: Suppose we have two populatos, 9

30 populato : Also, let sample : y, y,, y w, w,, w K N ad populato : K N.,,, K ad sample : z, z, K, z are draw from populato ad populato, respectvely. Let u y ad u w be the populato meas of populatos ad, respectvely. Let ad be the sample meas of samples ad, respectvely. z z The, the populato covarace s whle the sample covarace s N y y w w N σ yw, z z z z. z Itutvely, s z would be very large postve as the observatos two populato are larger or smaller tha the sample meas smultaeously. That s, the observatos are postvely correlated. O the other had, s z would be very small egatve as the observatos oe populato are larger tha the sample mea whle the oes the other populato are smaller tha the sample mea. Therefore, the observatos are egatve correlated. Fally, s z would be close to 0 as the observatos oe populato beg larger tha the sample mea whle the oes the other populato are sometmes larger but sometmes smaller tha the 0

31 sample mea,.e., the observatos the two populatos are ot correlated. Eample:. Let be the total moey spet o advertsemet for some product ad sales volume ut 000 packs. z be the z z z z z 99 z 0 0 s. Note: s z s ot scale varat. For eample, the above eample, f the sales volume s ut pack. The, z would be 5000, 5700, 400, 5400, 5400, 800, 600, 4800, 5900, Thus, s z wll be 00, whch 000 tmes larger tha the orgal oe. It s ot plausble sce the correlato betwee the total moey o advertsemet ad the sales volume would chage as the measuremet ut chages. The quatty troduced et s scale-varat ad ca be used to measure the correlato of two populatos. II Let Correlato Coeffcet: σ y : populato stadard devato for y, y,, y K N σ : populato stadard devato for w, w, K, wn w s : sample stadard devato for s z : sample stadard devato for,,, K z, z,, z K.

32 The, the populato correlato coeffcet s whle the sample correlato coeffcet s σ yw ρ yw, σ y σ w s s r z yw s. z Note: ρ ad r yw z Eample cotue: 0 z z s.4907 ad s z The, r z sz s s z 0 z 0 z 0 0 z z 0.9. Note: r z s scale-varat. For eample, eve the sales volume s measured pack per ut, the value of r z s stll the same, 0.9. Eample: Let z,,,,4, z

33 The,, z 6, s 5 5 z 5 5, s z z 5 0, 5 z z s z 5 5. Thus, r z s s z s z Note: whe there s a perfect postve lear relatoshp betwee varable ad z, the r. r mght dcate a postve lear relatoshp. z z Ole Eercse: Eercse 4.. Chapter 5 Itroducto to robablty 5.. Epermets, Coutg Rules, ad robabltes Epermet: ay process that geerates well-defed outcomes. Eample: Epermet Outcomes Toss a co Head, Tal Roll a dce,,, 4, 5, 6

34 lay a football game Ra tomorrow W, Lose, Te Ra, No ra Sample Space: the set of all epermetal outcomes, deoted by S Eample: Epermet Sample Space Toss a co S{Head, Tal} Roll a dce S{,,, 4, 5, 6} lay a football game S{W, Lose, Te} Ra tomorrow S{Ra, No ra} Coutg Rules: the rules for coutg the umber of the epermetal outcomes. We have the followg coutg rules: Multple Step Epermet: ermutatos Combatos. Multple Step Epermet: Eample: Step Step Epermetal Outcomes throw dce throw co T H,T,,H T H,T,,H T H,T,,H 4 T H 4,T,4,H 5 T H 5,T,5,H 6 T H 6,T,6,H 4

35 S {, T,, H,, T,, H,, T,, H,4, T,4, H,5, T,5, H,6, T,6, H} The total umber of epermetal outcomes 6 Coutg rule for multple step epermets: If there are k-steps a epermet whch there are possble outcomes o the frst step, possble outcomes o the secod step, ad so o, the the total umber of epermetal outcomes s gve by L. k. ermutatos: objects are to be selected from a set of N objects, where the order s mportat. Eample: Suppose we take balls from 5 balls,,,, 4 ad 5. The, two permutatos dfferet orders 5

36 Eample: N N- N N5 5 4 Eample: N N N N N- N! N- N- [N--] N! Coutg rule for permutato: As objects are take from N objects, the the total umber of permutatos s gve by 6

37 N N! N + N + LN N! where N! L N ad 0!.. Combatos: objects are to be selected from a set of N objects, order s ot mportat. where the Eample: combato, but 6 permutatos. Eample: C 0! combatos 0 permutatos 7

38 Eample: C 0! !6 combatos, total 0 combatos. Eample: 8

39 N N permutatos N N N- - C N! N!!!! 0! combato - - Coutg rule for combato: As objects are take from N objects, the the total umber of combatos s gve by 9

40 C Ole Eercse: Eercse 5.. Eercse 5.. N N!! N! N N! 5.. Evets ad Ther robablty Moder probablty theory: a probablty value that epresses our degree of belef that the epermetal outcome wll occur s specfed. Basc requremet for assgg probabltes:. Let e deote the th epermetal outcome ad e be ts probablty 0 e.. If there are epermetal outcomes, e, e, K, e, e + e + L + e Eample: Roll a far dce. Let e be the outcome the pot s. The, 0 e e e e4 e5 e6 6 Evet: a evet s a collecto set of sample pots epermetal outcomes. 40

41 Eample: E the evet that the pots are eve. E the evet that the pots are odd. E {,4,6 } ad E {,,5 } robablty of a evet: the probablty of ay evet s equal to the sum of the sample pots the evet. Eample: E {,4,6} e + e4 + e Note: S Ole Eercse: Eercse Some Basc Relatoshps of robablty c A : the complemet of A, the evet cotag all sample pots that are ot A. A B: the uo of A ad B, the evet cotag all sample pots belogg to A or B or Both. A B: the tersecto of A ad B, the evet cotag all sample pots belogg to both A ad B. Eample: E {,4,6}, E {,,5 } ad {,,} t E po s. The, E c E. E E eve pot s or pot s or both {,,,4,6 } 4

42 E E eve pot s ad pot s {} Note: two evets havg o sample pots commo s called mutually eclusve evets. That s, f A ad B are mutually eclusve evets, the A B φ empty evet Eample: E {,4,6}, E {,,5 } E ad E are mutually eclusve evets. Results:. A c A. If A ad B are mutually eclusve evets, the A B 0 ad A B A + B.. addto law For ay two evets A ad B, A B A + B A B [Ituto of addto law]: A B Ⅰ Ⅱ Ⅲ 4

43 ΙΙ A B, A Ι ΙΙ, Β ΙΙ ΙΙΙ A B I + II + III [ I + II ] + [ II + III ] I II + II III A B A + B A B II Eample: c c. E {,,5} {,4,6} E E. E E 0, E E E + E + 5. E E {,,,4,6}. We ca also use the addto law, the 6 E E E + E E E {,4,6} + {,,} {} + Ole Eercse: Eercse 5.. Eercse Codtoal robablty A B: evet A gve the codto that evet B has occurred. Eample: {} E : pot occurs gve that the pot s kow to be eve. A B: the codtoal probablty of A gve B as the evet B has 4

44 44 occurred, the chace of the evet A the occurs!! Formula of the codtoal probablty: B B A B A ad A B A A B. Eample: 6 {} {} {} E E E E Note: + B A B A c Note: A B A B A B B A Idepedet Evets: A B A or A ad B are depedet evets. B A B. Depedet Evets: A B A or A ad B are depedet evets.

45 B A B. Itutvely, f evets A ad B are depedet, the the chace of evet A occurrg s the same o matter whether evet B has occurred. That s, evet A occurrg s depedet of evet B occurrg. O the other had, f evets A ad B are depedet, the the chace of evet A occurrg gve that evet B has occurred wll be dfferet from the oe wth evet B ot occurrg. Eample: A: the evet of a polce offcer gettg promoto. M: the evet of a polce offcer beg ma. W: the evet of a polce offcer beg woma. A 0.7, A M 0., A W 0. 5 The above result mples the chace of a promoto kowg the caddate beg male s twce hgher tha the oe kowg the oe beg female. I addto, the chace of a promoto kowg the caddate beg female 0.5 s much lower tha the overall promoto rate 0.7. That s, the promoto evet A s depedet o the geder evet M or W. A promoto s related to the geder. Note: A B A B as evets A ad B are depedet. Ole Eercse: Eercse 5.4. Eercse

46 5.5. Bayes Theorem Eample : B: test postve A: o AIDS c B : test egatve From past eperece ad records, we kow A c : AIDS c A 0.99, B A 0.0, B A That s, we kow the probablty of a patet havg o AIDS, the codtoal probablty of test postve gve havg o AIDS wrog dagoss, ad the codtoal probablty of test postve gve havg AIDS correct dagoss. Our object s to fd A B,.e., we wat to kow the probablty of a patet havg ot AIDS eve kow that ths patet s test postve. Eample : A : the face of the compay beg good. A : the face of the compay beg O.K. A : the face of the compay beg bad. B : good face assessmet for the compay. B : O.K. face assessmet for the compay. B : bad face assessmet for the compay. From the past records, we kow A 0.5, A 0., A 0., B A 0.9, B A 0.05, B A

47 That s, we kow the chaces of the dfferet face stuatos of the compay ad the codtoal probabltes of the dfferet assessmets for the compay gve the face of the compay kow, for eample, B A 0. 9 dcates 90% chace of good face year of the compay has bee predcted correctly by the face assessmet. Our objectve s to obta the probablty A B,.e., the codtoal probablty that the face of the compay beg good the comg year gve that good face assessmet for the compay ths year. To fd the requred probablty the above two eamples, the followg Bayes s theorem ca be used. Bayes s Theorem two evets: A B A B A A B c c B A B A + A B A [Dervato of Bayes s theorem two evets]: A B B A B A c A c B A We wat to kow A B. Sce B ad B A A B A, 47

48 thus, c c c B B A + B A A B A + A B A B A B A A B A A B c c B B A + B A A B A + A B A c, Eample : A c A The, by Bayes s theorem, A B A 0.99*0.0 A B c c A B A + A B A 0.99* * 0.98 A patet wth test postve stll has hgh probablty of o AIDS. Bayes s Theorem geeral: Let A, A,, A K be mutually eclusve evets ad A A L A S, the A B A B B A B A... A B A + A B A + L+ A B A,, K,., [Dervato of Bayes s theorem geeral]: 48

49 49 Sce A B A A B, ad... A B A A B A A B A A B A B A B B L L, thus, A B A A B A A B A A B A B A B B A L Eample : * * * * A B A A B A A B A A B A B A A compay wth good face assessmet has very hgh probablty 0.95 of good face stuato the comg year. Ole Eercse: Eercse 5.5. B A B A B A A A.. A B A B A B A B B A B A B A

50 Chapter 6 robablty Dstrbuto 6.. Radom Varable Eample: Suppose we gamble a caso ad the possble result s as follows. Outcome Toke X Moey Y W 0 Lose Te 0 0 I ths eample, the sample space s S { W, Lose, Te}, cotag outcomes. X s the quatty represetg the toke obtaed or lose uder dfferet result whle Y s the oe represetg the moey obtaed or lost. I the above eample, X ad Y ca provde a umercal summary correspodg to the epermetal outcome. A formal defto for these umercal quattes s the followg. Defto radom varable: A radom varable s a umercal descrpto of the outcome of a epermet. 50

51 Eample: I the prevous eample, X: the radom varable represetg the toke obtaed or lose correspodg to dfferet outcomes. Y: the radom varable represetg the moey obtaed or lose correspodg to dfferet outcomes. X has possble values correspodg to outcomes X { W}, X{ Lose} 4, X{ Te} 0 Y has possble values correspodg to outcomes Y { W} 0, Y{ Lose} 40, Y{ Te} 0. Note that sce Y 0 X, Y{{ W} 0 0X{ W}, Y{ Lose} 40 0X{ Lose}, Y{ Te} 0 0X{ Te} That s, Y s 0 tmes of X uder all possble epermetal outcomes. There are two types of radom varables. They are: Dscrete radom varable: a quatty assumes ether a fte umber of values or a fte sequece of values, such as 0,,, K Cotuous radom varable: a quatty assumes ay umercal value a terval or collecto of tervals, such as tme, weght, dstace, ad temperature. 5

52 Eample: Let the sample space S { z z s the delay tme for a flght, 0 z }. Let Z be the radom varable represetg the delay flght tme, defed as Z { z t} Z{the flght tme s t} t,0 t. For eample, Z 0. 5 correspods to the outcome that the flght tme s 0.5 hour 0 mutes late. Ole Eercse: Eercse robablty Dstrbuto Defto probablty dstrbuto: a fucto descrbes how probabltes are dstrbuted over the values of the radom varable. I: Dscrete Radom Varable: Eample: Suppose the probablty for the outcomes the gamble eample s { W} { Lose} { Te} 6 6 X X X 4 0 Y Y 0 Y 0 40 Let f be some fucto correspodg to the probablty of the gamblg outcomes for radom varable X, defed as 5

53 f f f 4 0 X X X f s referred as the probablty dstrbuto of radom varable X. Smlarly, the probablty dstrbuto f y of radom varable Y s 6 f f f y y y Y Y Y Requred codtos for a dscrete probablty dstrbuto: Let a, a, K, a, K be all the possble values of the dscrete radom varable X. The, the requred codtos for f to be the dscrete probablty dstrbuto for X are a f a 0, for every. b f a f a + f a + L + f a + L Eample: I the gamblg eample, f s a dscrete probablty dstrbuto for the radom varable X sce a f 0, f 4 0, ad f0 0. b f + f 4 + f 0. 5

54 Smlarly, f y s also a dscrete probablty dstrbuto for the radom varable Y. Note: the dscrete probablty dstrbuto descrbes the probablty of a dscrete radom varable at dfferet values. II: Cotuous Radom Varable: For a cotuous radom varable, t s mpossble to assg a probablty to every umercal value sce there are ucoutable umber of values a terval. Istead, the probablty ca be assged to a small terval. The probablty desty fucto ca descrbe how the probablty dstrbutes the small terval. Eample: I the delay flght tme eample, suppose the probablty of beg late wth 0.5 hours s two tmes of the oe of beg late more tha 0.5 hour,.e., 0 Z 0.5 ad 0.5 < Z. The, the probablty desty fucto f for the radom varable Z s 4 f, 0 0.5; f, 0.5 <. f

55 The area correspodg to the terval s the probablty of the radom varable Z takg values ths terval. For eample, the probablty of the flght tme beg late wth 0.5 hour the radom varable Z takg value the terval [0,0.5]. s 4 Theflght tme beg late wth 0.5 hour 0 Z 0.5 f d *0.5. Smlarly, the probablty of the flght tme beg late more tha 0.5 hour the radom varable Z takg value the terval 0.5,]. s Theflght tme beg late moretha 0.5 hour 0.5 < Z f d * O the other had, If the probablty of beg late wth 0.5 hours s the same as the oe of beg late more tha 0.5 hour,.e., 0 Z < Z, the, the probablty desty fucto f for the radom varable Z s f, 0. Note that the probablty desty fucto correspods to the probablty of the radom varable takg values some terval. However, the probablty desty fucto evaluated at some value, ot lke the probablty dstrbuto, ca ot be used to descrbe the probablty of the radom varable Z takg ths value. Requred codtos for a cotuous probablty desty: Let the cotuous radom varable Z takg values [a,b]. The, the requred codtos for f to be the cotuous probablty dstrbuto for Z are a f 0, a b. b b a f d 55

56 Note: c d Z d c f d, a c d b. That s, the area uder the graph of f correspodg to a gve terval s the probablty of the radom varable Z takg value ths terval. Eample: I the flght tme eample, f s a dscrete probablty dstrbuto for the radom varable Z sce a f 0, 0. b d d. Smlarly, f s also a dscrete probablty dstrbuto for the radom varable Z. Ole Eercse: Eercse Epected Value ad Varace: I: Dscrete Radom Varable: a Epected Value: Eample: X: the radom varable represetg the pot of throwg a far dce. The, X f,,,,4,5,

57 Itutvely, the average pot of throwg a far dce s The epected value of the radom varable X s just the average, 6 E X f average pot. Formula for the epected value of a dscrete radom varable: Let a, a, K, a, K be all the possble values of the dscrete radom varable X ad f s the probablty dstrbuto. The, the epected value of the dscrete radom varable X s E X µ f a f a + a f a + L + a f a +L Eample: I the gamblg eample, the epected value of the radom varable X s E X f + 4 f f Therefore, o the average, the gambler wll lose for every bet. 6 Smlarly, the epected value of the radom varable Y s 0 E Y 0 f y f y f y b Varace: Eample: Suppose we wat to measure the varato of the radom varable X the dce eample. The, the square dstace betwee the values of X ad ts mea EX.5 57

58 ca be used,.e., be used. The average square dstace s ,.5,.5, 4.5, 5.5, ca Itutvely, large average square dstace mples the values of X scatter wdely The varace of the radom varable X s just the average square dstace the epected value of the square dstace. The varace for the dce eample s [ X E X ] Var X E E X the average square dstace f Formula for the varace of a dscrete radom varable: Let a, a, K, a, K be all the possble values of the dscrete radom varable X ad f s the probablty dstrbuto. Let µ EX be the epected value of X. The, the varace of the dscrete radom varable X s a µ [ E X ] Var X σ E X a f a + a µ f a µ f a + L + a µ f a +L Eample: I the gamblg eample, the varace of the radom varable X s Var X f + 6 f + 6 f Smlarly, the varace of the radom varable Y s 58

59 0 0 Var Y f y + 6 f y y f 0 II: Cotuous Radom Varable: a Epected Value: Eample: Z: the radom varable represetg the delay flght tme takg values [0,]. 0 Z < Z. The, the probablty desty fucto for Z s f, 0. Itutvely, sce there s equal chace for ay delay tme [0,], 0.5 hour seems to be a sesble estmate of the average delay tme. The epected value of the radom varable Z s just the average delay tme. E Z f d d averagedelay tme 0 0. Formula for the epected value of a cotuous radom varable: Let the cotuous radom varable X takg values [a,b] ad f s the probablty desty fucto. The, the epected value of 59

60 the cotuous radom varable X s E X µ f d b a. Eample: I the flght tme eample, suppose the probablty desty fucto for Z s 4 f, 0 0.5; f, 0.5 <. The, the epected value of the radom varable Z s 4 E Z f d d d Therefore, o the average, the flght tme s 5 hour. b Varace: Eample: Suppose we wat to measure the varato of the radom varable Z the flght tme eample. Suppose f s the probablty desty fucto for Z. The, the square dstace betwee the values of Z ad ts mea E Z ca be used,.e.,, 0 ca be used. The average square dstace s E Z f d The varace of the radom varable Z s just the average square dstace the epected value of the square dstace. The varace for the flght tme eample s 60

61 Var Z E Z [ Z E Z ] E the average square dstace. Formula for the varace of a cotuous radom varable: Let the cotuous radom varable X takg values [a,b] ad f s the probablty dstrbuto. Let EX µ be the epected value of X. The, the varace of the cotuous radom varable X s [ E X ] Var X σ E X u f d b a Eample: I the flght tme eample, suppose f s the probablty desty fucto for Z. The, the varace of the radom varable Z s Var Z E[ Z E Z ] f d d d Ole Eercse: Eercse 6.. Eercse 6.. Chapter 7 Dscrete robablty Dstrbuto 7.. The Bomal robablty Dstrbuto 6

62 Eample: X : represetg the umber of heads as flppg a far co twce. H : head T : tal. X 0 T T X 0 0 combato X H T T H X combatos X H H X combato X f umber of combatos the probablty of every combato, 0,,. X : represetg the umber of heads as flppg a far co tmes. X 0 T T T X 0 combato 0 6

63 H T T X T H T T T H X combatos H H T X H T H T H H X combatos X H H H X combato X umber of f combatos the probablty of every combato, 0,,,. X : represetg the umber of heads as flppg a far co tmes. The, X 0 T T.T X 0 0 combato 6

64 H T....T X T H T X M M O M combatos T T..H M M X f umber of combatos the probablty of every combato Note: the umber of combatos s equvalet to the umber of ways as drawg balls heads from balls flps. Eample: Z : represetg the umber of successes over trals. S : Success F : Falure Suppose the probablty of the success s whle the probablty of falure s. 64

65 65 The, 0 Z F F F Z combato S F F Z F S F Z combatos F F S S S F Z S F S Z combatos F S S Z S S S 0 0 Z combato combato every y of the probablt combatos of umber f Z,., 0,,

66 66 : Z represetg the umber of successes over trals. The, 0 Z F F.F Z combato S F...F Z F S F Z M M O M combatos F F.S M M combato every the probablty of combatos umber of f Z From the above eample, we readly descrbe the bomal epermet.

67 ropertes of Bomal Epermet X: represetg the umber of successes over depedet detcal trals. The probablty of a success a tral s p whle the probablty of a falure s -p. Bomal robablty Dstrbuto: Let X be the radom varable represetg the umber of successes of a Bomal epermet. The, the probablty dstrbuto fucto for X s! X f p p p K!! p, 0,,,,. ropertes of Bomal robablty Dstrbuto: A radom varable X has the bomal probablty dstrbuto f wth parameter p, the E X p ad Var X p p. [Dervato:] 67

68 E X 0 p p f! 0 p p 0 p p!!!!! p p p! [ ]!! j j p p j j 0 j! [ j]!! j j sce p p s j! [ j]! p p the probablty dstrbuto of a bomalradom varableover trals!!! p p! The dervato of Var X p p s left as eercse. How to obta the bomal probablty dstrbuto: a Usg table of Bomal dstrbuto. b Usg computer by some software, for eample, Ecel or Mtab. by some computg resource the teret, for eample, or Ole Eercse: Eercse 7.. Eercse The osso robablty Dstrbuto: ropertes of osso Epermet: 68

69 X : represetg the umber of occurreces a cotuous terval. µ : epected value of occurreces ths terval. The probablty of a occurrece s the same for ay two tervals of equal legth!! The epected value of occurreces a terval s proportoal to the legth of ths terval. The occurrece or ooccurrece ay terval s depedet of the occurrece or ooccurrece ay other terval. The probablty of two or more occurreces a very small terval s close to 0 osso robablty Dstrbuto: Let X be the radom varable represetg the umber of occurreces of a osso epermet some terval. The, the probablty dstrbuto fucto for X s where ad µ s some parameter. µ e µ X f, 0,,, K,! e.78 ropertes of osso robablty Dstrbuto: A radom varable X has the osso probablty dstrbuto f wth parameter µ, the E X µ the epected umber of occurrece s ad Var X µ. 69

70 The dervatos of the above propertes are smlar to the oes for the bomal radom varable ad are left as eercses. Eample: Suppose the average umber of car accdets o the hghway oe day s 4. What s the probablty of o car accdet oe day? What s the probablty of car accdece two days? [soluto:] It s sesble to use osso radom varable represetg the umber of car accdets o the hgh way. Let X represetg the umber of car accdets o the hgh way oe day. The, ad The, X f 4 e 4! E X 4. No car accdet oe day X 0 f, 0,,,K e ! 0 e Sce the average umber of car accdets oe day s 4, thus the average umber of car accdets two days should be 8. Let Y represet the umber of car accdets two days. The, ad The, Y f y 8 e 8! E Y 8., 0,,, K car accdets two days Y f y 8 e 8! 8e

71 Eample: Suppose the average umber of calls by 04 oe mute s. What s the probablty of 0 calls 5 mutes? [soluto]: Sce the average umber of calls by 04 oe mute s, thus the average umber of calls 5 mutes s 0. Let X represet the umber of calls 5 mutes. The, ad The, X f e 0! 0 E X 0., 0,,,K 0 0 e 0 0 calls 5 mutes X 0 f ! How to obta the osso probablty dstrbuto: c Usg table of osso dstrbuto. d Usg computer by some software, for eample, Ecel or Mtab. by some computg resource the teret, for eample, or Ole Eercse: Eercse 7.. Eercse 7.. 7

72 7.. The Hypergeometrc robablty Dstrbuto: Eample: Suppose there are 50 offcers, 0 female offcers ad 40 male offcers. Suppose 0 of them wll be promoted. Let X represet the umber of female promotos. The, X # of combatos for 0 female promoto # of combatos for 0 male promotos # of combatos for 0 promotos X 50 0 # of combatos for female promoto # of # of combatos for 0 promotos combatos for 9 male promotos X 50 0 # of combatos for female M promoto # of combatos for 0- male promotos # of combatos for 0 promotos X # of combatos for 0 female M promoto # of # of combatos for 0 promotos combatos for 0 male promotos Therefore, the probablty dstrbuto fucto for X s 7

73 X, 0,, K, Hypergeometrc robablty Dstrbuto: There are N elemets the populato, r elemets group ad the other N-r elemets group. Suppose we select elemets from the two groups ad the radom varable X represet the umber of elemets selected from group. The, the probablty dstrbuto fucto for X s r N r X f, 0 r. N r Note: s the umber of combatos as selectg elemets N r from group whle s the umber of combatos as N selectg - elemets from group. s the total umber of combatos as selectg elemets from the two groups whle r N r s the total umber of combatos as selectg ad - elemets from groups ad, respectvely. How to obta the hypergeometrc probablty dstrbuto: e Usg table of osso dstrbuto. f Usg computer by some software, for eample, Ecel or Mtab. by some computg resource the teret, for eample, 7

74 or Ole Eercse: Eercse 7.. Chapter 8 Cotuous robablty Desty 8.. The Uform robablty Desty: Eample: X: the radom varable represetg the flght tme from Tape to Kaohsug. Suppose the flght tme ca be ay value the terval from 0 to 50 mutes. That s, 0 X 50.. Questo: f the probablty of a flght tme wth ay tme terval s the same as the oe wth the other tme terval wth the same legth. The, what desty f s sesble for descrbg the probablty? Recall that the area uder the graph of f correspodg to ay terval s the probablty of the radom varable X takg values ths terval. Sce the probabltes of X takg values ay equal legth terval are the same, the the the areas uder the graph of f correspodg to ay equal legth terval are the same. Thus, f wll take the same value over ay equal legth area. For eample, wth oe mute terval, the 0 X f d X f d L 49 X 50 f d 0 Therefore, we have

75 f, 0 50; f 0, 0 otherwse Note: sce we kow f c some costat, the by the property that 50 0 f d 50 0 cd 0c c 0. I the above eample, the probablty desty has the same value the terval the radom varable takg value. Ths probablty desty s referred as the uform probablty desty fucto. Uform robablty Desty Fucto: A radom varable X takg values [a,b] has the uform probablty desty fucto f f f, a b; f 0, b a The graph of f s otherwse. f /b-a a b 75

76 ropertes of Uform robablty Desty Fucto: A radom varable X takg values [a,b] has the uform probablty desty fucto f, the [Dervato]: E X b + a, Var X b a b E X f d d b a b a a b a b a b a b+ a b+ a b b a b a a The dervato of b a Var X s left as a eercse. Eample: I the flght tme eample, b 50, a 0, the E X , Var X Ole Eercse: Eercse 8.. Eercse The Normal robablty Desty The ormal probablty desty, also called the Gaussa desty, mght be the most 76

77 commoly used probablty desty fucto statstcs. Normal robablty Desty Fucto: A radom varable X takg values [, ] has the ormal probablty desty fucto f f where f e σ µ, - σ, π µ E X, σ Var X, π.459 The graph of f s f u ropertes of Normal Desty Fucto: 77

78 a ad σ µ σ µ E X f d e d πσ µ f d µ µ σ Var X e d πσ b µ the mea of the ormal radom varable X the meda of the ormal radom varable X X the mode of the ormal probablty desty f u > f, µ µ X u c X s a radom varable wth the ormal desty fucto. X s deoted by X ~ N µ, σ d The stadard devato determe the wdth of the curve. The ormal desty wth larger stadard devato would be more dspersed tha the oe wth smaller stadard devato. I the followg graph, two ormal desty fuctos have the same meas but dfferet stadard devatos, oe s the sold le ad the other s the dotted le: f u 78

79 e The ormal desty s symmetrc wth respect to mea. That s, f u c f u + c, where c s ay umber f The probablty of a ormal radom varable follows the emprcal rule troduce prevously. That s, u σ µ σ µ σ X µ + σ % X µ + σ % X µ + σ %.e., the probablty of X takg values wth oe stadard devato s about 0.68, wth two stadard devatos about 0.95, ad wth three stadard devato about. Stadard Normal robablty Desty Fucto: A radom varable Z, takg values [, ] has the stadard ormal probablty desty fucto f f where f e, -, π µ E Z 0, σ Var Z. Note: we deote Z as Z ~ N 0, The probablty of Z takg values some terval ca be foud by the ormal table. The probablty of Z takg values [0,z], z 0, ca be obtaed by the ormal table. That s, 0 Z z the area of the rego betwee tw o vertcal les z 0 e π d 79