MATH-STATS 5312 Lecture Notes 8-30-17.pdf

MATH-STATS 5312 Lecture Notes 8-30-17.pdf - x02" Made...

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Unformatted text preview: x02" Made «(7 W7 7‘41 ”was“ :3 (W1)? Payday“: ,4 [97441617604 W'a' 6\ WM Mca'HAaEC-v n réfut. «#4 '4 W gum c/Mx/usvég. Arms/1c) = (Mam:— Diéf/H'It—w'h'u-e Lam: Av (Bnc) = (Adm/MAW) An(3uc)=mn6)umnc) 9.0. Manfrm Lw: (AUB)’= A71 13’ (Antsy =A’ui5’ A’=EI 3 r 7 7} 3’:{;—é 7X?M}' Probability We consider the probabilities of events in a random experiment. An experiment is a process of making an observation or taking a measurement. A random experiment is an experiment with uncertain outcome. An event describes something happening in a random experiment, it is a set consists of some elementary outcomes. ' Example: Tossing a coin is a random experiment. Obtaining a head is an event. We are uncertain whether we will get a head or a tail when we toss a coin. We want to find the probability of obtaining a head. Example: Measuring an individual’s height is a random experiment. We are uncertain what the individual’s height is before the measurement is finished. Height between 65 inches and 70 inches is an event. We may want to find the probability that an individual’s height is between 65 inches and 70 inches. Interpretations of Probability There are three interpretations of probability. 1. Classical Definition. If an experiment can terminate in N equally likely and mutually exclusive ways, and an event can happen in K of these ways, the probability of the event is KIN Example: Tossing a coin, the probability of getting a head is 1/2. We have N=2, K=1. Example: Rolling a die, the probability of getting a 2' or 5 is 1/3. Wehave N=6, K=2. 2. Frequency Theory. If the experiment can be repeated independently under the same condition, the probability of an event is the relative frequency of the occurrence of the event in an infinitely number of trials. Example: Tossing a thumb tack, what is the probability that it \i/ ’V will point up? We can find this probability by repeatedly tossing the thumb tack millions and millions of times. The relative frequency that it points up Will tend to the true probability. Example: People are either right handed or left handed. What is the probability that a randomly selected individual is left handed? We can find this probability by observing millions and millions of people. The relative frequency of left-handed people will tend to the true probability. 3. Subjective Probability. Subjective probability depends on the individual’s degree of belief. Example: Wéather man/woman reports the probability of raining is based on his/her personal assessment of the weather condition. Let E be an event. The probability of E is denoted by P(E). LfltgbefigW/fis’aaaWMcfif 74$“ P ,c'aa é ”luau-€145}; M fwds'm Cinfim£ o-r [’W'hlea, 6/ W) I. pcmzo «(fl/165. 2 P(S)= . 3 1+A01A’;A3. ' WW“ J3 and ”Le/‘14; =¢, 1:15;, %« Him: gm,» famfwA-ve P. A" 4&7tfivflv/k P 0A :3 (Ag) and»? ' £2 (P, ,.) P lama 1’72 7% FWWW A, P{A)=/—P(A’) PM: 32AM’ ‘ and Ang’: ¢ 5; FW‘lL-éa. (2.) and (3) (7(6): PMVA’) [=P{A)+P(,q’) ‘~ WA): I—P(A’) 7% I PM”) 30 PM: lfifl=¢,%u/V=§ 17(45):" P(§);/'/ 2.0 ”W fl AC3 m PM) 513(8) PW: BzAUUBfl/l') W’ AME/WW Balm/van? (f) P(B/IA’)20 ‘3 (3) 2 WW 5 Pm: P(Au(BnA’)) =P(A»)+.. NBAA') ‘ 2 pm) 44w P0403) =P(A)+ Hg), was) 17%: Wake A U3 = A um’n B) :3; WWW"; (3) was): WNW/1’05) * Ah» 5 = (An 8) UM’nB) Pm): P(AnB)+. P(A’n B) P0m BF PCB) — I’M/"3). swfihto. w *- poqu 5): PCAH PCB) - PC4013) +4“ PCAUB)=‘PO4)+ PU?) 5" AwfinoZZed¢orAam W4WWMW/LM, M‘Iahwwqflhj 4 m5. A={2, 4,5} 844,5} Ans-:ftr} PCA)=—§—=J5— I3 :1?- :J— P() 6 3 F(A/IB)=..Z'— --'L .L-..__4L-_§__. ?(AUB)— 11'3" ‘6'? c:[/, 1 4—}- AUBUc:{/,z,¢,s',£} A0344} Anc={* 4“} P04)+PLW+PCC)=%+f-'+g—=-§—’ 4 PWBHH/‘MCH' P(Bflc)= .64. i‘v-f-z- :2— P(A'nl3flC):.é-— = 8 ~i-r‘L 3'- LS:- PCAJBUC) 7 6 g 6 'l4 Tragwwrm ixpfww ané W 1:: 7;" ‘5’“«1’ 2; w M «7 w I) :6.ny 0? W flaw/6C: Mia/med»... W343 Adi. 11‘ 0,0,6; ”VB/wt, Pl ‘-'— PCC’)+P(62)+P(63) 102‘ P(C.ncz)+P(C.nc;\+ chzncg) PI”? 4%, NC. U C; u 6;) ’[Wfincm F(C.nca)+ m, = P(C.)*P(c.)+ We) 0(3)] Efxé' Wig; W M ’- MW%M‘czfl'm M 'I-I-WWAcMJ-‘t NV'VMW “"4me B W %a77//:;45~mw“;:. W14 Mao/B CM’Kaa/a/wafia'rnn “M?" M 2.. Ad/m‘m M RWCL 3. PWfiaw 0f rob 9'05ch ordereedfian- up; .7: I47 amfiur in (4.604 “anew/L; 4M¢ee I c = ”' .—.._..’3:_——- n Y (Y) Y'Uavr)’ 5km;- ‘4 V -MCY ’YPY u. n! . __ _______.. (Ia-r): hCY" r, "CY " err)’ m! __ h ) J, 44 M7,, ml: "4% ’u o . . This is sown-Hm Caflfid ”We’lm’m'dé'flmfuf 6,, u z m, ”‘7‘ 5» 9. W74 ’Tha +4; gal-234 W 5,?» “ 5‘." CMWW x“ 4", n, n, ~ 33" Ruflxz' Jw/‘i ...
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