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Lesson . Conditional Probability Review SA §z Dynamic and STochasTic Models Fall z§Ë Uhan Lesson h. CondiTional ProbabiliTy Review Ë JoinT disTribuTions ● LeT X and Y be random variables ● X and Y could be dependenT: for example, ○ X = service Time of ûrsT cusTomer in The shop ○ Y = delay (Time in queue) of second cusTomer in The shop ● If we wanTTo deTermine The probabiliTy of an evenTThaT depends boTh on X and Y, we need Their joinT disTribuTion ● _e joinT cdf of X and Y is: ○ NoTe: The TexTbook by Nelson uses Pr{X = x, Y = y} To mean Pr{X = x and Y = y} ● Suppose X and Y are discreTe random variables: Takes values , , ... ○ X aË az Takes values , , ... ○ Y bË bz ● _e joinT probabiliTy mass funcTion (pmf) of discreTe random variables X and Y is: ● We can obTain The marginal pmfs as follows: ● We can deûne a joinT probabiliTy densiTy funcTion (pdf) and marginal pdfs for conTinuous random variables in a similar fashion Ë Example Ë. _e Markov Company sells Three Types of replacemenT wheels and Two Types of bearings for in-line skaTes. Wheels and bearings musT be ordered as a seT, buT cusTomers can decide which combinaTion of wheel Type and bearing Type They wanT. LeT v and W be random variables ThaT represenTThe Type of bearing and wheel, respecTively, in replacemenT sets ordered in The fuTure. Based on hisTorical daTa, The company has deTermined The probabiliTy ThaT each wheels-bearings pair will be ordered: W Ë z h pvW Ë z/˧ Ë/˧ Ë/˧ v z Ë/z§ Ç/z§ h/z§ a. WhaT is Pr{v = Ë and W = z}? b. WhaT is Pr{v = Ë}? c. WhaT is Pr{W = z}? z Independence ● LeT’s consider events of The form {X ∈ A} and {Y ∈ B}, for example: ○ A = (þ, z6] ⇒ {X ∈ A} = ○ B = { , 6h} ⇒ {Y ∈ B} = ○ {X = a} can be wriTTen as {X ∈ A} wiTh ○ {Y ≤ b} can be wriTTen as {Y ∈ B} wiTh ● Two random variables X and Y are independenT if knowing The value of X does noT change The probabiliTy of Y (and vice versa) ● MaThemaTically speaking, X and Y are independenT if z Example z. Are The random variables v and W in Example Ë independent? Why or why not? h CondiTional probabiliTy ● CondiTional probabiliTy addresses The quesTion: How should we revise our probabiliTy sTaTements abouT Y given ThaT we have some knowledge of The value of X? ● _e condiTional probabiliTy ThaT Y Takes a value in B given ThaT X Takes a value in A is: ○ _e revised probabiliTy is The probabiliTy of The joinT evenT {Y ∈ B, X ∈ A} normalized by The probabiliTy of The condiTional evenT {X ∈ A} Example h. In Example Ë, whaT is The probabiliTy ThaT a cusTomer will order Type z wheels, given ThaT he or she orders Type Ë bearings? ● If X and Y are independenT, Then: ● LeT A ⊆ B. _en, if X and Y are perfecTly dependenT (i.e., X = Y), Then: h CondiTional disTribuTions and expecTaTions ● LeT X and Y be discreTe random variables In parTicular, suppose Takes on values , , ... ○ Y bË bz ● _e condiTional probabiliTy mass funcTion (pmf) of Y given X ∈ A is: ● _e condiTional cumulaTive disTribuTion funcTion (cdf) of Y given X ∈ A is: ● _e condiTional expecTed value of g(Y) given X ∈ A is: Example . In Example Ë, ûnd The condiTional pmf of W given ThaT v = Ë. Example þ. In Example Ë, suppose ThaTThe proûT from selling Type Ë, Type z, and Type h wheels is ¤ , ¤@, and ¤Ë§, respecTively. Find The expecTed proûT from wheels, given ThaTThe cusTomer ordered Type Ë bearings. þ Law of ToTal probabiliTy ● We can wriTe a joinT probabiliTy as The producT of a condiTional probabiliTy and a marginal probabiliTy: ● Using This, we can also decompose a marginal probabiliTy inTo The products of condiTional and marginal probabiliTies _e law of ToTal probabiliTy. Suppose is a discreTe random variable Taking values , , .... _en: ● X aË az ● We have a similar law when X is a conTinuous random variable Example @. In Example Ë, The condiTional pmf of W given ThaT v = z is: b Ë z h Ë/Ëz Ç/Ëz h/Ëz pWSv=z(b) Use This wiTh your answer To Example To ûnd Pr{W = z}. þ Example 6. LeT Y be a random variable ThaT models The Time for you To geT promoTed from O-h To O- in years. _e cdf of Y is ⎧ − Ë a ⎪Ë − þ if ≥ § ( ) = ⎨ e a FY a ⎪§ oTherwise ⎩⎪ (NoTe This is The cdf of an exponenTial random variable wiTh mean þ). a. WhaT is The probabiliTy ThaT promoTion Takes longer Than þ years, given ThaT iT has already been h years? b. WhaT is The probabiliTy ThaT promoTion Takes less Than @ years, given ThaT iT has already been years? @ @ Exercises Problem Ë. Professor I. M. RighT oen has his facts wrong. LeT X be a random variable ThaT represents The number of quesTions he is asked during one class, and leT Y be The number of quesTions ThaT he answers incorrecTly during one class. _e joinT pmf of and is: pXY X Y Y § Ë z pXY Ë Ë/h §§ X z Ë/ Ë/Ëz § h h/Ë@ Ë/Ç Ë/ Ç a. WhaT is The probabiliTy ThaT Professor RighT answers all quesTions correcTly during one class? b. WhaT is The probabiliTy ThaT Professor RighT answers Ë quesTion incorrecTly during one class, given ThaT he is asked Two quesTions? c. Explain why Ë, z §. pXY ( ) = Problem z. _e Simplex Company uses Three machines To produce a large baTch of similar manufacTured iTems. z§% of The iTems were produced by machine Ë, h§% by machine z, and þ§% by machine h. In addiTion, Ë% of The iTems produced by machine Ë are defecTive, z% by machine z are defecTive, and h% by machine h are defecTive. Suppose you selecT Ë iTem aT random from The enTire baTch. a. Deûne The random variable as The machine used ( Ë, z, h ) To produce This iTem. WriTe The pmf of . M M ∈ { } pM M b. Deûne anoTher random variable D ThaT is equal To Ë if This iTem is defecTive, and § oTherwise. Find The probabiliTy ThaT D = Ë given M = m, for m = Ë, z, h. c. Find The probabiliTy ThaT D = Ë;ThaT is, The probabiliTy ThaTThe randomly selecTed iTem is defecTive. Problem h. Simplex Pizza sells pizza (of course) and muõns (ThaT’s weird). LeT Z and M be random variables ThaT represenTThe number of pizzas and muõns in one order, respecTively. Based on hisTorical daTa, The company has deTermined The joinT pmf for and : pZM Z M M § Ë z pZM § §§.§ §.§@ Ë §.zþ §.ËË §.§þ Z z §.˧§.§Ç §.§6 h §.§Ç §.§6 §.§ a. WhaT is The condiTional pmf of M, given ThaT Z = z? b. WhaT is The expecTed number of muõns in an order, given ThaT iT conTains z pizzas? c. IT Turns ouT ThaT Pr{M = Ë} = §.hþ and Pr{M = Ë S Z = h} ≈ §.h@Ç. Based on This informaTion, are M and Z independent? Why or why not? 6.
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