Probapility cheet sheet, Cheat Sheet of Mathematics

Download Probapility cheet sheet and more Cheat Sheet Mathematics in PDF only on Docsity! e ia called mutually CLclusive oy, 1 Ms L nrg if they can never NAPPEN Simultaneously Mr ony kb i; AORS g. — * x A coll llection of events ig hale lo be CX haw f if a atleast One of the Coll Han re 2are to i occur in every performan eo sondom <xperim ent te Collec tian Say KET GY I& exhaw re pan, raha sS KEL t the Car rei Pong d *) 0 <P) <) 4) FO) ~) 2 PCB) 4 Y PC) ~1-pG 9) D Teted Tetal Rokability Troremt ef A, } Ar, Re; ahs VAR Ore Palrwive mu hua lly Cv clusive event , thin probabj lity of- weg. POA tA at = £9) = POY) 4 Pr ee) 4 Xlo racic def ™ Pr APR it called Probability fu } thy UNnigud Numbers PAH) Conespondsng to any event AEG Wy called thy probobility of the event A if tho fellow ing Axioms axe Soka fed: Anion) > Pin>o Ly any event AEA AxiornGi POPS) A xiore Gi) Ai pfy>.--,4 . be courtebly infta’ le No: of palvwise miially' 2 e xclulive evente yes MAHA Perit) AME D How PORTA ois wc ddligset -) =P) + Play )+--- On) (& ADR I Sobabi lity Space Deductions DPMAIPY 2) P(W=0 DrPH y AACE > pC <P? . S) FO+@) = PO)+PLD — PUP BD E) Leduction of clos tical def” 7 Continuity theorem rot Probab irhy o~ At IF Fan} be ~ mrnalate Sequence ob evenig , thon p(Lim An) = Lim PAn) y indepen dort: Lee sal } viv! a ee said to be ml tually independe nt fy p9D = = PAD VEO =P PO pte) PCW r(P= vy prc) = P00) UD PCS Nett a tually in dypendnie => Fatouoise dndipen der t. = py G24 Converse 76 net true Sandown vasioblee Arval yalued Linction XS POR Ke colled a random - voriable if for every seal number x the set fees: t= <K (8) EX GEA aes ds anevent: Samp/e Point in’. Speetsum of the Sandom Variable X2 XD: Te range of the Sondom jontobole x Fe cabbd the Jpecluum of X. : ae let ve lists ibu fon finebi on dor om Vaviokle " ROD PCwbexex) Vr er. X Trany Neal numbes. F, Rs To, | Broperticss 2 ¢ x Pa monotony ne reatin g £ 3) PlaexeL). Fy lb) ~ Fy fa) 0 Sab cori fates pi 5) Left cont » x @)-~ foo. plx=a) 6) Ke) =4, Reso Vigne Set of Prins of clitconts atmost countably. 2) Fy Can have On] n chon ‘ nuity of iy y pump discontiny py (no dis Continuihieg of urd Kind) | a OO Discutt ¢ Cases _ a . ) Spectr of X= x) =n : (az +/ ¢tlp ‘ “4 9) } {robobili om Mass Fanction p (Perm. f Je oh By (x) = pn P=) , fy 1 € [pecrum of y eliewhere. * IPG. 4 LESpectru mX * £@) >So VxE Spe clrum xX OR) f@so ver PRPC eX 2s SPO) [ * ELpectyum | = > f OD and Xt ex Y Plcxch) =~ > Lx ‘ Oe Heb i ate 5) At “A non- Secteur point “a” J PUza)= a 6) Foy uw C Spectrum x PO: x) = f (x) =f} = F,Ca)- 5a? Jump at point % eee ee ettgpeec ee Re. ee ea ee Som edn ostant Contnunut chistyib butiong 4) Untfornn Diststh button s- Xv Unifirm (a,b) Spee vu or ~ (4,4) £, (x) ~ a # O2reu, = 0 , eltewhore X~) Nm, ) (mes Fedametorg, o>o) Spectrum of x = Ce .0) e x Oe) = owt lo Po MY eK 3 Standerd Normal dittributiont 7 xn No, ) - 6 } ar | five Ale) = ae pO Meee vi 5) = 6O) ey Bat. t) Conchy ¢ distribution 5 woe (anchy (A,4) , Avo GU a wo A Poramerty , a. f@ a ae 7 ~ 6 OU 2x5 5) Gamma distributions Xn 32) Z Loo = O - elsewbe re. 6) ‘Noha Kota distribution of oe Kind x Bilt)» L map £-) - LG) » ~ 7 oD aM Gaal, Orig 9 ? elewhore » Leta Za distrlbution of Qf, fe Coe Ln >O fi Gero d By Fass ae y OLA ER » €ltevbore . Visson distribution a *A Lom? ly of Sanden variables x (we te T4 which dtpends porametrically or Hme t i Called a sto chaste PHO cose. Vef” No-af- ch - . 7 On Fd Of Stocay tie proces ma Given interval Lobe fying belo tro Lows fe Hows portren dist vibutfen . 4) Te noo chanye dlusing the Hime intral CH tth) indbpendont of no-af changy sccured in (ot) , Lr all + § ACO & Ws l)% Probab} lid of exactly and change ny C4t4b) 1 Piven by dh+ oth) whove ol ig A +e constant € ofh) th a funchorn of. h St LM 55 og L>D GW) Tho Probability of More in C1 44b) 18 Oth), han Ore chong XG): Noof changee in inksvad (2,4) P(OxG) =i) = zee a7 plrohu,.. “a a Ler @ of POISOry Procely Cng ‘moot changes perunit tiny) At Avg noof Changee INO Gilry Hime trterved (0,0 i - Propertiee_ Ob Expeclations p E(a)> Qy A> lonclan! 2) E[a-go)] Sg [9¢x)] 4 > (ony tan] 3) E [9,60 49,0)4-.-4 Intr)) = €(9,Cd)+ € (64) threes +€In(y) 4) le (xJ) < E[ I9cx3/) S) FP 90) 20 jth elgii>o #9 4f IM20 and EL 9OI}~0, tory gen Me; the pry IX) hak a one point abstributh ~ atg (2) 0 Qo oper ties of Vortance $ ——- Va5(x) ~ EL Cx-me 9 4 Var(x)> 0 => X =My , er the whole Probability Mabe ig Concentroded of the moan, ® Ke Quantity LG A>My . 3) Var X)= Car Cx) Wt) gt Ebb-m? > ELy2d “ony Gi) ae Ed x(x-D~ Mx (ny ~) i ~a4 1@ minimuny avhon Bett ye Re ot 5) Var @xtb) = a’ vas (6) srk Ae ~ Standondized Fandom Variables X bea &Vv with mean- Mx Standard oleviatten 760) Tropertiose | 2) ECV=0 2) Voslx* Je 4 2) Ureful for Comparing Af Corent AC vibubeny eee | Momenté =~ D th Order Gntrol moment My= EX Ox-m IG 5 pe 0/12, 08 Mp4, My >4, Aly > oe? 2) sth Srdby Nav Moment H, > EXx™) REE Ko= 4 oy =m, ? 3) yth moment about a Sve point la” G= ELonaty af rt exiet ¥20/1,2 5 > | Moma pl genesatin fnchont i (m-g: f) Pre vided expeetabirs Oy My GQ) = cL etXg in ambh of G4, 4) 4>0_ 1) Mxlo) always eactrts oe 2) My G) may G1) may not exis r 5 ZY MYM) = =F ~E OK) eee De soil ER 4) “ral > ae MCh) ? es +0(-®) Ss) XY ave vue having Of ly Mx (t) ond My Q) ves pect ively. Thy X ¢ Y, | have the Same Probabi I; ty Lit Hribuhen 4 ond anly if MCD = My €) ido FicaLly eeag anol Kastor? . )) 3,5 Meature of ASymaytyy . *t) Hoy +) re Roh Nery van) Shing Odd e@rdhy df 2) I. Meda Of Aint of Gist Yi Ea hitny iy >t ae "fy, = Fo —?