概率论与数理统计英文Word文档格式.docx
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Sincetheoutcomesofarandomexperimentcannotbepredictedinadvance,theexactvalueofarandomvariablecannotbepredictedbeforetheexperiment,wecanonlydiscusstheprobabilitythatittakessomevalueorthevaluesinsomesubsetofR.
2.Distributionfunction
Definition3.1.2Letbearandomvariableonthesamplespace.Thenthefunction
.
iscalledthedistributionfunctionof
NoteThedistributionfunctionisdefinedonrealnumbers,notonsamplespace.
Example3.1.3Letbethenumberwegetfromtossingafairdie.Thenthedistributionfunctionofis(Figure3.1.1)
Figure3.1.1ThedistributionfunctioninExample3.1.3
3.Properties
Thedistributionfunctionofarandomvariablehasthefollowingproperties:
(1)isnon-decreasing.
Infact,if,thentheeventisasubsetoftheevent,thus
(2),
.
(3)Forany,.Thisistosay,thedistributionfunctionofarandomvariableisrightcontinuous.
Example3.1.4LetbethelifeofautomotivepartsproducedbycompanyA,assumethedistributionfunctionofis(inhours)
Find,.
Solution
Bydefinition,
Question:
Whataretheprobabilitiesand?
Example3.1.5Aplayertossestwofairdice,ifthetotalnumbershownis6ormore,theplayerwins$1,otherwiseloses$1.Letbetheamountwon,findthedistributionfunctionof.
Letbethetotalnumbershown,thentheeventscontainssamplepoints,.Thus
And
so
Thus
Figure3.1.2ThedistributionfunctioninExample3.1.5
Thedistributionfunctionofrandomvariablesisaconnectionbetween
probabilityandcalculus.Bymeansofdistributionfunction,themaintoolsincalculus,suchasseries,integralsareusedtosolveprobabilityandstatisticsproblems.
3.2DiscreteRandomVariables离散型随机变量
Inthisbook,westudytwokindsofrandomvariables.
Definition3.2.1Arandomvariableiscalledadiscreterandomvariable,ifittakesvaluesfromafinitesetor,asetwhoseelementscanbewrittenasasequence
Assumeadiscreterandomvariabletakesvaluesfromtheset.Let
(3.2.1)
Thenwehave,.
theprobabilitydistributionofthediscreterandomvariable(概率分布)
X
a1
a2
an
probability
p1
p2
pn
注意随机变量X的分布所满足的条件
(1)Pi≥0
(2)P1+P2+…+Pn=1
离散型分布函数
Andthedistributionfunctionofisgivenby
(3.2.2)
Ingeneral,itismoreconvenienttouse(3.2.1)insteadof(3.2.2).Equation(3.2.1)iscalledtheprobabilitydistributionofthediscreterandomvariable.
Example1
Foranexperimentinwhichacoinistossedthreetimes(or3coinsaretossedonce),constructthedistributionofX.(LetXdenotethenumberofheadoccurrence)
n=3,p=1/2
Xpr
11/8
23/8
33/8
41/8
Example2
在一次试验中,事件A发生的概率为p,不发生的概率为1-p,用X=0表示事件A没有发生,X=1表示事件A发生,求X的分布。
two-pointdistribution(两点分布)
X
1
P
1-p
p
某学生参加考试得5分的概率是p,X表示他首次得5分的考试次数,求X的分布。
geometricdistribution(几何分布)
X
2
3
4
…
k
q1p
q2p
q3p
qk-1p
Example3(射击5发子弹)
某射手有5发子弹,射一次命中率为0.9,如果命中目标就停止射击,如果不命中则一直射到子弹用尽,求耗用子弹数x的概率分布。
*****
Example3.2.1
Adieistossed,bywedenotethenumbershown,Assumethattheprobabilityisproportionalto,.Findtheprobabilitydistributionof.
SolutionAssumethat
constant,.
Sincetheevents,aremutuallyexclusiveandtheirunionisthecertainevent,i.e.,thesamplespace,wehave
thus.Theprobabilitydistributionofis(Figure3.2.1)
.
Figure3.2.1ProbabilitydistributioninExample3.2.1
Question.Whatisthedifferencebetweendistributionfunctionsandprobabilitydistributions
例2有一种验血新方法:
把k个人的血混在一起进行化验,如果结果是阴性,那么对这k个人只作一次检验就够了,如果结果是阳性,那么必须对这k个人再逐个分别化验,这时k个人共需作k+1次检验。
假设对所有人来说,化验是阳性反应的概率为p,而且这些人反映是独立的。
设表示每个人需要化验的次数,求的分布(constructthedistributionof)
Binomialdistribution(二项分布)
Example3.2.2Afairdieistossed4times.Letbethenumberofsixgot.Findtheprobabilitydistributionof.
Solution.Thepossiblevaluesofare.
Firstwefindthe