CH.25 Discrete Random Variables

Transcription

1 CH.25 Discrete Radom Variables 25B PG #1, 3, 4, 6 25C PG #1, 3, 5, 8, 10, 11 25D PG #1, 3, 6 25E PG #1, 2, 3, 7, 10 25F.1 PG #2, 3, 5 25F.2 PG #2, 4, 7, 8 25F.3 PG.804 #2 25G PG #2, 3, 4, 6 Review Set 25A PG # Review Set 25B PG # Review Set 25C PG # 1

2 25ADiscreteRadomVariables Aradomvariablerepresetsiumberformthepossibleoutcomeswhichcouldoccurforsome radomexperimet. AdiscreteradomvariableXhasasetofdistictpossiblevalues.Ithiscourseyouwillcosider olyafiiteumberofoutcomes,sowelabelthemx 1,x 2,x 3,...,x. EX : Theumberofbicyclessoldbyabikeshop.Theumberofhouseswitha3Acargarage. AcotiuousradomvariableXcouldtakepossiblevaluesisomeitervalotheumberlie. EX : theheightsofme,whichwouldlieoaiterval50cm < X < 250cm. thevolumeofwateriaraiwatertakwhichcouldlieotheiterval0 < X <100m 3. 25BDiscreteProbabilityDistributios IfXisaradomvariablewithsamplespace { x 1,x 2,x 3,...,x }adcorrespodigprobabilities { p 1,p 2,p 3,...,p }sothatp(x = x i )= p i,i = 1,...,,the: 0 p i 1foralli = 1to p i = p 1 + p 2 + p p = 1 i=1 { p 1,p 2,p 3,...,p }describestheprobabilitydistributioofx Notatio Statemet P(X = 3) theprobabilitythatxequals3 P(X 3) theprobabilitythatxisatleast3 P(3 < X 7) theprobabilitythatxismoretha3butomoretha7 EX#1: Statethemodeadmediaofthedistributio x P(X = x) probability EX#2: Showthatthefollowigareprobabilitydistributiofuctios a) P(x)= 2x ,x = 1,2,3,4b) P(x)= 3 x x spike graph ( ) x ( 0.4) 3 x,x = 0,1,2,3 2

3 25C Expectatio Iftherearetrialsofaexperimet,adaevethasprobabilitypofoccurigieachofthetrials, thetheumberoftimesweexpecttheevettooccurisp. TheexpectedoutcomefortheradomvariableXisthemearesultµ.Theexpectatioofthe radomvariablexis E(X)= µ = x i p i or x i P(X = x i ) i=1 i=1 EX#1: Irolladie54times.Howmay3'sdoIexpecttoroll? EX#2: Fidthemeaofthedata. x P(X = x) FairGames SupposeXrepresetsthegaiofaplayerforeachgame.ThegameisfairifE(X) = 0. EX#3: Iagameofchace,aplayerspisasquarespierlabelled1,2,3,4.Theplayerwisthe amoutofmoeyshowithetablealogside,depedigowhichumbercomesup.determie: a) theexpectedreturforoespiofthespier Number Wiigs $1 $3 $7 $13 b) theexpectedgaioftheplayerifitcosts$8toplayeachgame c) whetheryouwouldrecommedplayigthisgame. 3

4 25DVariaceadStadardDeviatio Ifadiscreteradomvariablehaskpossiblevaluesx 1,x 2,x 3,...,x k. withprobabilitiesp 1,p 2,p 3,...,p k. the thepopulatiomeaorexpectatioise(x)= µ = x i p i thepopulatiovariaceisvar(x)=σ 2 = x i µ ( ) 2 pi = E(X µ) 2 thepopulatiostadarddeviatioisσ = ( x i µ ) 2 pi EX#1: Fidthestadarddeviatioforthedistributio: x P(X = x) E Properties of E(X) ad Var(X) Properties of E(X) If E(X) is the expected value of the radom variable X, the: E(k) = k for ay costat k i.e. E(5) = 5 E(kX) = ke(x) for ay costat k i.e. E(3X) = 3E(X) E(A(X) + B(X)) = E(A(X)) + E(B(X)) i.e. E(X 2 + 2X + 3) = E(X 2 ) + 2E(X) + 3 for fuctios A ad B. Properties of Var(X) Var(X) = E(X 2 ) (E(X)) 2 or Var(X) = E(X 2 ) µ 2 Properties of E(aX + b) ad Var(aX + b) E(aX + b) = ae(x) + b ad Var(aX + b) = a 2 Var(X) EX#1: Fid: Suppose X has the probability distributio: x p x a) the mea of X b) the variace of X c) the stadard deviatio of X 4

5 EX#2: Xisdistributedwithmea7.4adstadarddeviatio2.15.IfY = 3X +5,fidthe meaadstadarddeviatioofthey=distributio. 25F The Biomial Distributio Biomial Probability Distributio 8 EX#1: a) Expad b) A basketball player makes 80% of his three throws. If he shoots 4 free throws, determie the chace of makig: i) two shots oly ii) at most three shots The Biomial Probability Distributio Fuctio Cosider a biomial experimet for which p is the probability of a success ad 1 p is the probability of a failure. If there are idepedet trials the the probability that there are r successes ad - r failures is P(X = r) = r pr 1 p ( ) r, where r = 0, 1, 2, 3, 4,...,. P(X = r) is the biomial probability distributio fuctio. The expected or mea outcome of the experimet is µ = E(X) = p. If X is the radom variable of a biomial experimet with parameters ad p, the we write X ~ B(, p) where reads "is distributed as" EX#1: 5

6 EX#1: 86%ofuiomembersareifavorofacertaichagetotheircoditiosofemploymet. Aradomsampleoffivemembersistake.Fid: a) theprobabilitythatthreemembersareifavorofthechageicoditios P(X = 3)= b) theprobabilitythatatleastthreemembersareifavorofthechagedcoditios P(X 3)= c) theexpectedumberofmembersithesamplethatareifavorofthechage. E(X)= 25F.3TheMeaadStadardDeviatioofaBiomialDistributio SupposeXisabiomialradomvariablewithparametersadp,soX B(,p). ThemeaofXisµ = p. ThestadarddeviatioofXisσ = ThevariaceofXisσ 2 = p 1 p EX#1: ( ) p( 1 p) Afairdieisrolled18timesadXistheumberoffivesthatcouldresult.Fidthemea adstadarddeviatioofthexadistributio. 6

7 25G The Poisso Distributio Whereas the biomial distributio B(, p) is used to determie the probability of obtaiig a certai umber of successes i a give umber of idepedet trials, the Poisso distributio is used to determie the probability of obtaiig a certai umber of successes withi a certai iterval (of time or space) Examples are : the umber of icomig telephoe calls per hour the umber of misprits o a typical page of a book the umber of fish caught i a large lake per day the umber of car accidets o a give road per moth. The probability distributio fuctio for the discrete Poisso radom variable is P(x) = P(X = x) = mx e m for x = 0, 1, 2, 3, 4, 5,... x where m is the parameter of the distributio. If X is a Poisso discrete radom variable with parameter µ the µ = m ad σ 2 = m. If X is a Poisso radom variable the we write X Po(m) where m = µ = σ 2. Coditios for a distributio to be Poisso : 1) The average umber of occureces µ is costat for each iterval. It should be equally likely that the evet occurs i oe specific iterval as i ay other. 2) The probability of more tha oe occurece i a give iterval is very small. The typical umber of occureces i a give iterval should be much less tha is theoretically possible, say about 10% or less. 3) The umber of occureces i disjoit itervals are idepedet of each other. EX#1: Betwee9:00amad9:15amoFridays,Sve'sFloristShopreceivestheXdistributio ofphoecallsshow a) FidthemeaoftheXGdistributio X Frequecy b) ComparetheactualdatawiththatgeeratedbyaPoissomodel. 7