Poe bali pudta, on Pa pwatsatnd Saunple Space: A Beto} poral se dwt be Totstag a Cota Totstng 98 Coto, Taw. v4 atl ce Vi aa, te Of thoovoty ade § ae Prr{ermane tr a reamdona eayeten ow Event: A non- tonpty Subnet @ Sanne Kate. Exhauative vents An woe Conarsttng a, ww lhe vOrehous, Powibiltia 4 Catted an eirhoussive ain. Be. Ta tossing O Coto Math exelurive Two G saose wds CHL Mutual Crhocumttve Wels Said +o be nkuat ly exclusive prevent tbe ctmuttemeous “hay Jrod amd fost age mutual ak % teod a , A the haapprelng q one iweat praia a th olhind. fx* In tosstug a Coin, ayetttng Pxilugive tn view 1 ibe ou “4 fot & vol porattole Ale wants Comnel be fo omotlos than Such Abe tuan outs Gert Equatiy Meet events! Th one a Expected to happen tm prehrrene pont pseghows to be teal Ledy.St: Tu Aoeating Q Coxn, “teow G Ake head @ the tofi uy Gually Led. bebelity : Pobabittiy : Ly the oukgome aa Lviad Conatiia "! ethautsive, waudastdy erelurive , Squat ‘poast ble cages, ah, winieh on a thea ase fovoratle Casey to am event &- oer precousttty o% “happening tke wok € 4 ee n Nuwles dA possible Cases the probably Com at sont be Guat +0 4, because te Jovorable (aaa aud te number 4 rot eae other. ee posstole Cones Cam at wot Coiuctde Th a & probabity % not Hogting & © te, hagpenting 4 E then dy wvaleay pe) + PE) = => praet: Exo ples ! ) the probabtly quiteg head’ tm tossing | = “the porstole gudtoes ase “poss the (enchoustve J corer (0) = a hang awd toh: Nuwher Nuwbhs 4 jovousable case (mm) =! 1 Dpobobitey a qerrg hal CP) = Te» the Peoboilty oy qe ca) & numbys qveadlin tam 9. tb) om odd number > Nuvker oj porstble Cases | guttomnes (a) =6 (© Nuwloer of fowurable suttomes ek, age owousscle do Hee ever Sluce 3, ey 6 6 -Turmabtas Probate itty oO geuy a sumbsr qpocatew greater thaw 9 Cp) = #2 Cb) Numbes af fovouralsbe uk Comes (on) = 3, cone Yovouratle to lhe evn giwe b 3 \ preobabtttly quit oad number (DW =2 =e Ce) Stetlady we hove tp) = 2h, Shute & & 6 ame Jowourable fo eo dune _ ety Additton ramen 1 prooaksltty puting a one OF Shake ment ‘ ae othe a other wauk watt Sum oy te tf A, & age Hoo waka pla o 8) = PW + Le,-s00{ ate Proof fet la otak number of grchaurtive, meuduabty, Prelustve and equatly possible Cases In the tvtats be n. out q tage Let omy Cary be ferousable do the wink A curd “ma Cag be Jovouratle do 8. Henee dhe viumbts GQ Crs fovouranle +0 ether Ao & Awe) - meme | ms Phere) = MEME. M4 HE —3O Stace TH Case ane Fovourable tok, PA) = EE Sluice ma Cas axe forvousuole to 8 P@= ma Submitting disse tn te ns 4 ® we gr PCR or B= PLA)+ pce) this provy le additon dhscrm Of potoability Axtows g Probably mio ute Sample Spore awd & us the Set | ast eves E. We Quecfate a unique Keak number p= pla) Frown le profoaki lity Oy be Went A PL Ake following oretorns One Sorbtafted w) plo=l Wwe) 4a wey Toe 4A ee Cw Th AD Ra, Asie Ate Oe wautualiy e xtlusive AyUAg.- An) = PCD PCAS) + + 9(An) dor +o each event A fa 0 2He! evints a E diem pC6 . : Prove ok pC AUB) = Peale PCE) — PLANE), whine A ond one at un duets. Powoh Casecty 24 A Gd B OMe wauduaiiy excletive events Nese, Angad thm pane) =0 (ym nn 9 pty) Cased) + TL A od B one omy 0? wos. Huse, ANG > 7 . Ls @-A => ave = Avls-A) ANB = pCave) = Pp (eves) AS RA OHH drajotaks (uduatry erndusve ) BA = B-(ANB) > pce-a) = PB) - PCANe) = pw + ple => PCALB= P(A) + PB) - plans) pave) = PChd+ PCa) - PLArB) fn Ge 9 wurhuatty TecluaVe Wks them Note WI} Ay, fas - 4p {pceoe.) = PlAhav-- QW TL A, @ & 6 aoe ou thee PCA) + PCB) + PCO = PCANB) — P (Bac) — pcan +? (Anzac) @) - OBhn) = PCPa) CAR) > + 9(A) 2 wit, then P @oBvd =D A box Conlain, 2 whtte, 5 black avd 6 ved baw. TL a boll u drawn ak vondem cdrod & Alo proloabt Hy jhat th Lo Either sed @ whe # Sol. Tolod mumber a bal inthe bow = iy. p C gene Sau) Chamiey) ol the x) 7 P Ca etttag a wlite bot -2 b Cgqerkug 0 ee sty Hee equrfrecl probability 4 Uy, a A bexr-t Contain a rite and @ black bath ond a Q White avd & lolack boats. TE me Cath woes hot uthe Colour * Second loox Conteins boat & drroron oF Famdem from Probodetlity sfrok Hoy ose Ate Aarme % bey wm te Airdt box = QW4 2B= uy So: —Tofak Mumaloers Toad newer a baly Mw da S2Cond bot = QWEUB=6 Case @) : Suppehe brortts “He bails oe valdte » tum tbo 7 22 paddalot ly u —* 2. + Suppose both tle poy aou bloek, htm the 60: u 2 ao probab Cty u E x 2S Rate etlher GL Hose tro Casy ase Joveusatlle +0 te event, dle wequixed poobabt ity Vere Ya —® - . The b prebatsitiny thot a Gludeat pasted 0 pliyter 1 ny 8 Zand dhe pockabitty hat dos pase belt o phapics 4 tad : aye ool UL nthe probability bed He perso Least ene beg Be. what tu the probability ‘ha4 Ss We poems Ale English bot ? SOP. As le Studete pana a pheysics test Bi le gludat pasa @ Euelith tal. aa Pass pans) = a penve= & Pple=? By Additon Jaw of psobolotlty PCALB) = pray + P(B~- plans) = 2 _ Sh Se. genet > prem. + - By, eo . ue ee > [eres = ane tossed ff © -ro dice a dle be probe A qetng au tela A 8. t le Space Soh: In a epamdoen tots H Guo Atce Songle Se by 9,324,664 * \ Us wm(c)= 6xb= 36 g, 3,4, 616 Su 3%, S65A Cyetttng Ou even maumbes on dla teen die = 4 auch x Ay as & 3 65 (A= 3x6 = 18 ptay = 0 . 1 ot mas) as 2 RB: tle Sum aq Ke Poier on Hla two Atte U &. Be § @o, (69, BIS Ke) acess pca)= ae - & ns) ac Al‘o ANB = 1 (a,c) aA”, (es us} - AANR) = 3 pang) = sats) as \ Now pequived peobabs try u pave) = PCA) + PCB) ~ pcace) = Wy S.-3 = Boe SsAddittom ow of probabilny ; { W PpCav® = Pea) « PIB) - png) ae D pCaue = pra + PO ThE A,B ane tn0 mutually” Gulustve wets @ AFB We wurkualey Atagolut. Multiplication Jao of proloalot Hy : plans) = PCH. P(B/A) (t) pCans) = play. Pls) Th Oh aud B aH (pane) = PCA): PCR) fr dependent weds Conditenak probabtlity ¢ kek K and & be te Hee events. Proloalet ty Q tee happening q Hie covet @ taken te ewe ATS! Crvendtiy heppied! happened is Catted toe Conditional prbabitty deoted yp (BIA). plela) & the probability of 8 Y uty 4 dhe occunente OL both B dud A fvin Ae Probalt P (BIA) = Probability the ocesence OY dle given Qwik fe fe, plBfay = PUAOB) ayo plarg) = POAC PLA) Re,4 Bayes fitosem on Conctfional povbabstry Sdokernet fat Ay, Ap, An be @ See ql exhaustive wots o the Sample Space $ ond Mudusly exclusive rth PCAt) to for each Om | olhes work asotard wth Mm, (AC Om) vsti CA) to then at P UtcJa) = BEAD) PIAL) é pear) P Mar) Prod. ite Rave $= AUARU-- VAn owel Acs A= SAA = (Avaav.-. DA) ah Using debiousve Larwo whe Rus we bave . (Anda) Stace AAA tes TI fon aoe mutually exclusive. pe hove boy agpiying, the addition vube A prodabtlsty. PCA) = P CAMA) + PCARIA)* autliplicalton cule orto Az (AOA) U@ana) O-- + PCANAA) Now Applying cath tysrm inthe Rus we twwe, PCA) = PCAD PIAIA) + plAa) p(AL Aa): °° + P(An)? (AIAN) chek &, PCAs S. PCANP MIA) @ 4q=1Me Condifenad probalsttty of At tor omy t give Ay wo defined by Pea) = PLMOAL «pean P(A He) — Pla) PCA) Uatng @ Qa thy demominates a le RAS we have, pletla) = PLM) AALAND Oy & 7000) val a) = this pros Raye’ teaeeen hen Condition potty —_———— Note: AM W PD) =0 Wd PA) = t= PCA) rdw A Be Complinunk A A Pardlolowns : oo | Uk STE Ar @ we dap wks honing Uae: a awd pland)> Vie» Conagrke the jee w plrss) td PEIN csi) PIB) re) Sl? pipig) = PAs) _ a rane). Me = B 9 is * pcpin . PBN . Me Lk ee ptr) — pin) = 2eaew pies AeBul Us PTB) = I- pep) = 1- (Ya) = % ie hove De- Mergen’ Lavo (Rue) = AOE a (Raw) = p(Aue) = 1- 7? Ave) = 1- fw + Me) - PNB] LO, plR as) = I1- (e+ Ya Yu) 2 PiRig) = PCAN®) . Sie s 18) % | * thus pCR IB) = S/e Ao p(B/a) = P Baa) _ shiz Sfuce Pra) “Ya Pact PIA) PWIA) = Sle a eee @ Tra Schel ach q ite Stuslenks failed tr {tH Nonguage, Ist @ Hla Stoduelt failed tm Second Language ndleuts patted & both. Tha tual nt ond es iy Actecte ak giao find Hie probattl a) he fated fe Hoe Janquage 1 fe tod fatled tu Hla Setend angen senna fonpnge he be totes ty) he fasted fn fe an Honk Jogeng®itt MAD Ne fatled tm efter Othe fie So: fk WW be dhe Sel qQ Stuolenls fail te the tient domquage ode ae See aed fasted te fla Setond Language. toe oe by ie PLW = 28 = a ee ise oo PLB? es, PAg) = foe 4 b> ® pare) = 2008) _ le. x no) 3/a0 @) PBA = pane) Vio : oe Peay Ve GH) -P ve) = PCA) + PLB) - PLAAR) Pfave) " eee be 1 Io Oothree machtas Ar B awd C proditce ouspeetively Gor, Zot, § lot a the cole runbes 07 tems occteny wthe percentag a qa del cetive output q Huse mrachinys A asap etiveltt ai, at and et. ganders amd found defective, find le probate Hy flock lie fem WOs produced by woking © An ttum & Suecked atX Sal: So tee A, B,C Sland for tle wets q Selection of au tem from machtne Ar B ¢ ale $2 20. = 32 co = 10 con. PCA) en OO PCB)= ao Oe PCc)= =O Suppose “BB Ha wut a Sdeoton YO defective Hem ua. 8 P (DIa) = = 008, p(D/s) = 455 =° 08, P (Die) = H= ook. loo a Selected Qe nd He probability rot woe need to Hud U(C/0) t sia woulsue Go Te +8 poodsced from we howe by Bayes cose ple) pried p(cio) = pea). POIA) + PCB) p(d/s) + Pw). POI) (ort) (0-04) (0.6) (0.02) + (0-3) (0-63) +(0:') (0.04%) thus 9 (e1d)= ols@1 BG lei factory He dou four warhfus Ar B C+D Wan. {aetutag Tespectively 20%. isthe, ast, Hop OQ the +ota? v ocks Production. out oy frese Sih, Kr, 3h, gtr one Athectve Ta bore drown af Santon was fourd defective robot 4 “ha probably tat 108s wamugatused by Ao Dt Gol ey date A, BoD wanupockrore Sof, 15%, asi and 40H Of ue dolar production. Huue we hwe pcnpso plays 02, PCB)= 1S = ols, pce) =028, 0 Lue “we be the Cunt qf Selecton of O defective Loelt shim _s aa PO&/A) = aE 2008, pix/g) = eke Pole) = 8-008, Pico) = B= eR toe eed to Compute P (ave Ix) we howe, Stace A Oud D OMe wkuat enolusive P (aud/x) = pla) + 9 O/x) Baye’ # Hheosesn PA). P&/A Pam PAIX) = oe ee Pla. PCRIA) + pte). PUB) + Re) Pec) 4 pep Pel?) toe have oy 1 as plalx) 2 02 ® 01 0F 0.8 KOS + ONS % 00% 0.85 % 0:03 + 0-4 R008shy Pls) = _O% 2 o3i9s 0. 031S Alo Plole) = pep Pb !P) © u) (0.08) 0.0315 ersecicia a oasl bey, p vn /*) = panes oack = OSHS has yw Secrebassts thamdding auuspectively Q the fila Q ow govecesoent epost. The prolcaatlty thot hey Wishile Gur Teport He orl, acl 6:08 find the preebraty blared on He first @ An office Qo, Corh, 16 aud sth gesgeetivety, O.0S, O-), cok He nifagi led suport Cau be Seovetaxy. Ars! ONKS qoronuta (A [E) = Prad ple /ay) pea) PIE/M) + PCA) PIE/Aa) + plas) PEIAs) + pls) pe /Au) OA Stadert A Cow Solve 45h A, Hn prblews given fa tay book awd 2 efudert & Con Solve todl-. volo 4 fie probability sok ethos Ao QB Com Sole g A problem Chosen ad Ran gan dow Ans: O0.4asq Rowden vantable To Teak variable is assoctale Tea Sandon Expestmont , both wvery eudcornd dun Ue WU Catted a Aaudor vastable. Random Variables one Usually leoroled fou XY 2° rot tte volur 1 Suppose ER? olde tossing O Coin, Wrad? auc 0 for te cteted for te outCorme ft Sasa! by Han gumdorn vortable Ww asso (2 Space @uttome tail. se tuth owt HO drum x WW) 24 ad XE) =0- Pompe 4 xe fo, % . DPscvete aud Contivirerss rorioon vartables * Disevere Ce loka fut @ Counntady cated & dt crete fufiutte sumboes Hordes, Vortable. duvole yartable Hotll namgqe LE WToMIng & coin amd ebsertug la out tome, at) Toss Wty Toastasy 0 Cotas and observing dhe rumbes of head cLusucting up Iho gordon Varkalle tak on Courtable tngintte nawber of value Shon Hh G Cae & Won- discrete @ Conttrucus gaudemn varia e, A Continuous Queda variableCorn essere amy value tn the enlesvas oO} oUt netmboss, fx: Hae tylak OL cverl¥cles. clonevto ste potwlus on a Speedomelys . Discrete — proehabslely Atitsflutton : ce eee! oe TL] fos each Volue Xt OL & dtsveete Nomdora Variable ch -lbak ¥, We amign a out number p%t), Su @) Plax) Zo an < be) =! wthun ae fumetion po Yu cated a poebabilty functton, Ty te protoalat ty stot % take elt Values 21 4 Pio pk ead she @ POW Ale Set oy values Xr, Pou] 4 Caled 0 Jsvaete (Yratte) prob ality dtybi button q the discwele Hondeorn vasiatle x. ile function pls) 4 Cabted the prcoabot Ute density ureston (P44) @ the paca th waa function (p. mm.) a dibtributton untst on 4) depts boy 4H) = pix s2) = & Kx), % bung an frteger qe) digtetloutive fut en (ed awe —lhen is Cartel the Cumula tye ti, He deftattens oh we ase abuaty ourtltar 9 (az) awd Vartante (=) fora querat ireaen e_ Edler ®) Aravatuten, that Hogs EBM go Zh Th mlnWe Cam 05 woe deftur die menu avd vartance a be to Ate evete probalet uy Atsetbuston, Tn ab Care boex) Cewscspondter 10 Si aud toe hove SACs SF pen as. “tle mean and Vartante ole dPusete proloalstlty Attrtbutfon U def deffued of follows nee oe cu) 7 Nartance (Y= So be-p). pees) t Standand deviakten (=) = VV Holes vartante Can also be pak tn fa tr 2 ve & RE pees) — [Erbe] Pe ‘oblewn4 D. A Cofn & tossed voice. A otamdow vartable X wepoesink tha number of heads tring up. find fia dtsowete probabsltty distetbution for X- Aldo pnd ty votam ond Voustaace So: Se PH, HT TH, TT, qu asoctaiton oy re slomunrts QS to te Darders Natale K are cw espeettvdy 8,1 0 Now pC&R) = < pad et, peep, POVELP(x =0, te, m0 ead ) = Prt) =e Ple=t, fe, me 1 bead) = per UTH) = PRT) + POW) = My te oa, P(eee, Ce, ahd) = pad= Zr 4 dtstxibublen jor kK UU follow, phe dteusete poobabttt Mean (NW) = xt P Oi) = OF teh = KrtKet Nostamte (Y= SOG - 4)? bee) 2 = o-et ate he @ue bed shy, we have mam =4 and Variance = fy == pottoutng Attribution vepoesortd 2 escaete @® Shoo fat tte fend diy roean od Varsiamnce. probability Atstrtlouticn . x lo Qo 30 leo Pe | Ye | 3a | 2a | % poe olobenve Trak a) >0- for abt “x aud S plo) 21. S pod: x = lo 4 Qo0-.3 3 a - ee a Sot Mecva (p) = B= lo + Gor Qvteo . 900 =2°5 8 3Nawtiance (NV) = § &- )® oo Q = tonash d wpe g + (eee + 3 2 (40 -as) “t = goby 5 gt 25 Sy as3 + gest 3 € V = G00 = z co thes Mum |W) =@s and Vartomee (V) = #5 D Feud the Volue oy & Such Arak foe fetteutng distribution crepreamts a fintte probablitty droetoutten. Bente Hed Ghd watan and Stardsod devtatfon- Als Hed pCxED> b wr) and b W478). x -3 -Q cl ° \ Q 3 dO) k ak 3k kk Bk ak ok Sol: joe must hove poo 2° for au x cieen ig Sodtayied OK SO oud dre Seterd Ke aks ke ERASE HEOKEK ST S lek=l °° Tee Hel gpobabt cq desiotbution U as I Q 3 jhe desesete| tentte Jolicuch xX -3 -2Q -) ° po Me She Fhe ‘te Fhe Me KeMean (WW) = Sx ho0 = “ye bP vont oF So pro Nartamte NW) = SL pw pe PGP LAs OF 2 2 2 2 —(-3). -2). 2 -). 3 Ue = (3) nicecaciee ie +) rot Gz) e 2 +a) & 3 . (ome 16 _ uo 2S aie fareasror arent = Fe 2 \6 shes, ket mew co. & Standand deviation le, oie VSR Also, ob (x21) = pea) + pGadt pe + Pts 6M ety, S2yphey SR 2B we *te* te +40 +e te Pen = pla + pe = 2abl. 3 16 Ne Te pU ” \ Ly &)-- (- oe") paar’ mn _ Oe eR) C8) EY ao xc ght . i. 4 Ne Buk NP em? a= (=p) = W m) -- 3) gee 3eel” pee ek} e aw man OF0 Jurthn % x , a =(ep) Si Aerated b->e. Aldo te actos Z odors OR). (= 2) St ana Thuy, v0e qu Graterbutien a the sls & known 23 Te potdien standom vastaile. pix) & alse cated persion probaly Aunctton amd ew UW Carted patton voxtoke. shu for Ws Ol 9, 3,-- was fhe sta dthtet buen prsdabtlt Ls © \ 2 3 - 7 7 arm pw e™ lc ¢ tt at 7 loc howe paz & oo 7 0 -89 5 SPW ema ME mee me ub a} ay ence PW uaVastanve (V) = 5. % POO ye 0 =o x Now Se Pp = 5 fecxd +3) pe 3) Oy E me wre wey (x-a)! tx)! x2 ” Sam eet ae. eo [ix mu pata jem _ 2 mm Om ow ¢.e%*™ ny © ations ni, pe@ pe hve, " ge, Sx ptr) Vatag This nasutk oo Nustamce (0) 2 en ek em = SD(D= VWwev™ Meam amd Standoad Deviation q “he potzsen Disbibuton..-Problewm < eee » plex a Cota ch exactly one ty tosked b ties. Hod fe prvbabitiy q qeteg haaat (71) abrrost 3 Meads C1) cxtleast & heads Gras 3 mek. Sol, Sol, Ae pe RNs Wer Os > cucceny Ue BM joe know shat he probalslty ‘a! x ta gue by piws= Nx prqn WD pCrbeady = ple) = tec, tous) los) = 0-ac p(x=o) 4 Pex + plxeart plxed) (9) P Cat moa 3 heads) = 2 Pe Hegos) (OD) = bey tostlos) + 4 Hog losyod 2 os)” (eu 6+ = 0-934 l- P (z-O+ per) t jj les ie ee, (0-8) (0-8) &¢, (0:5): (Rep Catdrast @ heads) = = O68tS —_—_ sty Arak 0 pm wenerfactusoal bag & fatory TR Suhr pers ane womurfetusd , brat G7 uaetly 2 084 defective 4) nore Y-bowe ae defeotve. D-le Probab be defective B a, tod U4 the probabtety C1) attest 2 a9 clefective ct Ag Ci) 02301 () 9 3KI (KI 0 Og reeDIwa quia Cente Q emntioeatog ‘Yes! @'No Whal Y “he Prooabrltly o quimivg auideast 6 ansilad Cormectty out 10 Questions asieed? Alte find the probabiltty othe Baume 4 sve ae & epttons for 2 Covseed Ansosrt« Sol Yee 0° devote fe Covsect Answer aud We have fr te 4rvst Case, p= Ye > 47 o-% lo pls Aly prgr™ 2 Mex (ye) (U5) = (x (A) re Asve 0 fad P (x20) - — lo lo lo lo lo p(xz6) = 30 Cet yt “Cet G + co | = t by uss = So [2 (DDFS all (i 4 vq Ay Jorma] = 0-377 Tr He Gelond Shaws Care volun He ase 4 options pe Ye. 45 Bly 2 NF10 poor = cq Yay Cle) = +b" rc, | pence plape) = led + pea) + Pes) + p (a) + Pte) ao i 7 . oc. x! - 76 [2 log + 3 cats cg +B gt Ge = > [fixate py xorg ace 3xior] i “ P(r) 9 ae QWD 4eve dice rere Abamon 96 Lime aud ve wuuabes a 4 me) an odd vuwbes actually uuid aul tr the erpertenerst a given ft a Btwowslad Aepptbution 10 Thr dete and Cabeedate te enpect frequindts. No. oy dice shoo 0 1 2 3 & c loa So " obsenwed heweny we | so. = prbabtlty Q qesng 18 assert “V4 pose doe % — 7 kimu au odd uber Lundy ouk wt Pmwe= Nox be 4 rdw neo powce POD = O40 oe Cae L Sx 3s ta a He binornkal prbesatcry diabdtaction perition. Srepert ed sprequeaciy ore Hrroem 46 then} Lashes on) Weare 7% treme YC)= > Sc,73 te spelt (Hearted) 4) = 3 Sc, = aneels 4(r) = 3 S(3 = 3B ¥iw* 30 prequensy aed Se ee 3,18, 303 \s,3 4-3: _—— wre > Sou Lreyeige OCs 2 3xl=Ou r Cotvs ane tossed l are - Lem ond the follaotng sesusts : a binomial dtotetlation Yew He dade ond Codewlode le Heovekcad prequarct “ewe % 0 1 oe & peu aq 36 as S POS The sequlned pamnesical fpequeeneses OO 4, 26, 22,266 6 FA a porAsen Areteibulion oe Hoe otlocoting dato aud Colowtate the Prrosekicod Fey. x © \ aS & ee Co SO pye Shalt feetot Comnpute tue enews (PP) ate gum ines batt on - bes si 5 ot bor so OFF pe st 122+ 6041S 4941 rye howe ener (pe or the por Aten ratot bation. te potéson deetet ution os wat og CA ADE 200" Poo Os pcwe= xs a qe, Y= 090- (os) € xt OS Bub 20-6068SO) = labs les” ae uw tr 4am ~e delat “Ce ~Heovettca) putting x20, |) & 3 Foequncta , “ey ase 08 fotlewos. ies (os)? onc)! ne ©S) 61, 6 lal.3 (o-s)* 3 ee sis, la3 OD 43, 13 ost a EN rm frequemct4 One thats , the ququbeed Hrrcseecad lat, 6 Is, 3.0 setorded MO u Gees Fite tu theosettcal $) the wemrbes acetdowks por day (2) 4 fertile dudessto nn & posted oH 00 doup petason Groper button qor He dota aud Caterlate fronueme lS: ~ o ' @ 3 & S 4 ops tee 37 18S \ rcs etical prequencta ane 123, 143, 56 IS 3,0 ay. te Date vawles F arctdonts Pra year te ani ceive fa a ctiy qpttews o poteson dtabetbation wate Sr 3. ouk % twwately ue vloys @G the Joos tox asvos 4nd aainesh acctdert (? qe with © 7° > acctdak eam ®) wre hour Sat. By dota, Treen (HL) = 3 > ky have Yor tte petmen digtetoubon P= ™m=3.“te pottten désivibutton i am gir by pone we” ett xt - at Lez 4(x)= (000 P&) = > $= \o00 we? x = 50. 3— Ste 6 P2005 as x oth no acetdenk tra Yor 1) Nuwbes a] dsivess 2 Hlo)2 G0 2 250. ol (8) prebatott mose thaw 3 acefdunk ma 4 vr ay peat 1p (x89) el [po pinr Ka * a] Le, o} i as 3 = [- [o.os (143 e bs ws) = 0-38 nunkes 4 dewuss out {ooo votth menvose thom Ma qe 3 acetdenks = lo0o x 0F8E = 380 —=Continuous probobttty fumttien : el fee evey “ot lockouging 4o the sense a4 Continuous Mowdom vastolle K, We asigqn Aaokeiiy he de Conditions tH) 40020 Ct) {300 2\ “ atttiey quwetion dhs foo Y Carlet 2 cone nu9s PP 4 % ton (P44) @ padoalottity Aas 4 edt a Keak numb 40 Y @b) & fie Subtutenol Q fe damage space O * then hs proboabiliy ret % By te (ob) UY Segusd 0 be THe awd b- Gwe $09 between @ em ORGQ es a $Ooda a awd Vortoble rola protools Hh > BG & Cukinu0 unezion FOO deptord by uastty 4umetion 4oy ton he Fo = PO 4X) = Fieve —b we Cabted te Cumudatve didtrtouson quuesion C44) q x. Noles ay ca a omy Maat wane ben a @ Plz = s Yludx + 1- play) \- ( 4 Qode 7v y p az © Pte) & pas) ”Mean avcl Vasdante ee csaebescica a stardom Variable roth Prcloaleility cere, the men (Pp) yx ba Condi uous dunsity fue tion Yoo lw @ expectation ECO and lhe Vasiomte (Vax 4 Aeyuud a5 follows ~ {a foods bo Meow we) = vasiance (a8) = j (x- pe 409) ax wate Yudveduce too Continuo pobalet Gly Athteioutions we NOW cw (ea orenod aR Mawwely dla exprewtial Atdkiouton “Ee ponemtiod pistebusionS =e eats tion howto Tee Conti rnous probably Axa density dunetion Yoo qies ey = { ke** for %70 © otes0the whi &>0° we known 0s te exporersod deatel bation. tose awd we Aove . vided Faodee fee™ ole = [2 ML cede) —k -b Thus i fodus b - be bye tools Le Condi Rood cseqpsoed fora don Sart Ay | Probabiltty duwssty fueron. Coctinucns prebabiity tuncionOud Shavdord Deriokon o} tle mene DU GAS to bo Meow () = | aehods = \ x. he de ‘be ped fp adn Aprlytos, Reser? & out y oe “4 p- ‘Pe (SS) (4 at (neve XX -y0 as Xoo ae u tow! et a pa< Pe wee a? ol paris we how wus) Vastamce (o-®) = Jiep: fo dx ~b —®- of. fe p" on dan : woe ones Appying Bownoulleg aie 18 n ae + @ el [e-PS -ae-p[ rc +8 -« felewh BPO ewf- 56-9]ee “Thx Jor the Expoumbiad daveioation Maam () = + soviet Novena “Diakstbution : le Cowtinusns probably geatibubion hotug Hae pect dowstty {uastton 49 ques b4 fy 2 ae (e- py /o# o VaR rahee -0 < X20 >» Viewer didstsloution . Evidently $2) 20 00 < pee aud w>0 B kmovon 03 fre to re ; j 40) doe = : { eo (a-pY/ae* 4, bo o yan a puting t- CF @ 4: parm et, bochae dxaVied var & abe vats 5 —o te te Pe yunce [ 4lade = = fet prod -e olan 2 my bo - tern [adres ae® | eo dt wn od ° 4 wd “Pay = va qwuma famcon & 5 e Roy 4& [4 da = Le OS |. iA Q ; oles! fea. sts 400 mopeds O probousitny d button * awd Stamdand Deviation ats Kew Mum rr porn be? a mun pls J tiodt = —— jee os a VOR te have dr-ido-4 po Zee Bee = @ wt pric t we oe pasit Ua ee safe Og a Mam v \ 3 > wa & > > - a \a a QQ + Since oe Vasiante Va oe =—_——feeleen = ot v find ditch of la Jplloroteg fumesions Uo prrbatttnty cluasity, funtion, M 4,@e= { QX, 04%) olhesvoise G0 tag 2 ax, 44x! C ottres ide o fore J soodr= a Sof + SH Conditions for a peded oe @) earty fase bo 1 ! \ ( 4.00 = [ oe E [ese e f°], o) bo . © foo = oe gab oT Ct) he qs junseion Cou be asttten an tle feos Qx, | exeo talW = Qx, oe xe! ° oles os Tn A<%eo, aye an Y Lex thowx Boro to 1 \ fuses i a f Soe oe] 20 dat 20 AX = AL = I. ob 4 Roll He Conditions ase “el gastohed bade nota Pabb aeO ARud the Comglant “Such thot ene yee 04743 oO Otheavofhe A PAL. Algo Compute (f) p (14x48) ) pee Ui) Pfxsd (hy Moam OH) Vartan’ to SI™ femso He Kr Aldo He us hone { peoe! 3 fey feo dae => P| a1 =>9kel 2 kel Q = eel) 2 2 2 Wp (rexea) = wda = Ee 7 | atf{g- as (renee) {ie {atone 2) sale da \ 1 ; \ ! Gi) ptree [sonar = [Ha {z) oo 4 ot), at ° ° 3 3 ad 2 Blea a fo |e a) pas!) 2 f sors jee =] ns 1 3 wr pins) = |B 2° 3 3 3 edn [> good = fee aE gne | Bae 2 7D 7 36 — © Oo sled & to 36 3 W) Vartante =V = fae g at & Zax -( Pie fa® aor 3 ph = Jat Xo) 02 . x * SH - 3 a —l _ 38 al be ~ wot -_l Is le “3° ¢ 2 —= ® A *amdosn Voxtobole “gt has ha eae PO = / kek _gcxes ° AlRe treo. “| desta fonction Evaluate Ko omd fend ® plisxsa) Ci) piesa) GQ) p@a) Avs: el ket 2 , W® pQsxea) = + ay plxse) =o Wy) poy = & ® the es Klometse oun On drsusandds A Kem) notthou ously Sosk A Problem Im wespeck oO a Cestaim velida & o Tandem Vastable Roma Pad t —Xheo tims) © 3 x20 ° 5 X40 Akad the pooootilty Hnok 2 Vebscle is treclsh 4-vouble Hee C1) atleast gs 25000 k. mh QTY) aboot fos Aveo &-md GIR) betwerm (Gere 40 BaQo00 ky =Sat: apf Sele Nese KR & the Bander Vasioble sepomting Klesnetee Tr swattiple tooo Texting rouble ree sea toy Abe : Nelacte- a to ed PCR 2as) P Sl ? (xeas) as -% /yo € dx = I- J we 2 P(x pag) = 08283 ———. aid Wo fwd Pp Cx as) as Hix : peers) = f ty é ae as aalyo _ oe = 0 UCP Se (- é a — o cat) 5 3° \ athe 4 Mw a. aber x ' ko 6 29 Me Ale . -"] s-e +é le = 9-28) —2) Tod & om Crowned Variade roth mem 3 fad a) plod GI) PCx-Bd So: Hla Pidk a te Paeportertiod ‘Atta busin & given by %@ = { tee oexee ° otvoutte wte ence of tris dtotstbutton 4 qe by Wk By doko » mean = Yee 3 anh -e% Ye @ ane trente wo { ae ©, o-thesust Ke 7 M) Piso = |- plxe¥) if \ 3 = 1- f sends el- jee dx 0 VIS 0) tone ade (Ose 3 Wp (x23) = f acodeDT WY an exponential Varhoke YON "wngan S 8" O p(oer ei) at) p(-me% <10) CRY) PCx so ae AB: ct) PCoexel) = OT8l3 AN) p(-moenany = 0-860 CM) PCxeo @ x1) = or aie