概率论与数理统计英文.docx
- 文档编号:29295166
- 上传时间:2023-07-22
- 格式:DOCX
- 页数:27
- 大小:216.33KB
概率论与数理统计英文.docx
《概率论与数理统计英文.docx》由会员分享,可在线阅读,更多相关《概率论与数理统计英文.docx(27页珍藏版)》请在冰豆网上搜索。
概率论与数理统计英文
4.ContinuousRandomVariable连续型随机变量
Continuousrandomvariablesappearwhenwedealwillquantitiesthataremeasuredonacontinuousscale.Forinstanee,whenwemeasurethespeedofacar,theamountofalcoholinaperson'sblood,thetensilestrengthofnewalloy.
Weshalllearnhowtodetermineandworkwithprobabilitiesrelatingtocontinuousrandomvariablesinthischapter.Weshallintroducetotheconceptoftheprobabilitydensityfunction.
4.1ContinuousRandomVariable
1.Definition
Definition4.1.1Afunctionf(x)definedon(-〜:
)iscalledaprobabilitydensity
function(概率密度函数)if:
(i)f(x)_0foranyxR;
oO
(ii)f(x)isintergrable(可积的)on(-〜:
:
)andf(x)dx=1.
-nO
Definition4.1.2
Letf(x)beaprobabilitydensityfunction.IfXisarandomvariablehavingdistributionfunction
x
F(x)二P(X乞x)二f(t)dt,(4.1.1)
_oQ
thenXiscalledacontinuousrandomvariablehavingdensityfunctionf(x).Inthiscase,
X2
P(m:
:
X:
:
x2)=f(t)dt.(4.1.2)
Xi
2.几何意义
()
x
F(x)二P(X^x)二P((X,Y)|Xzx,0乞丫乞f(X))=f(t)dt
-oO
x2
P(x,:
:
:
X:
:
:
x2)=f(t)dt
x1
3.Note
Inmostapplications,f(x)iseithercontinuousorpiecewisecontinuoushavingatmostfinitelymanydiscontinuities.
Note1ForarandomvariableX,wehaveadistributionfunction.IfXisdiscrete,ithasaprobabilitydistribution.IfXiscontinuous,ithasaprobabilitydensityfunction.
Note2LetXbeacontinuousrandomvariable,thenforanyrealnumberx,
P(X=x)=0.0
P(a乞X乞b)二P(a乞X:
:
:
b)二P(a:
:
X
:
X:
:
:
b)
4.Example
Example4.1.2
Findksothatthefollowingcanserveastheprobabilitydensityofacontinuousrandomvariable:
kf(xrFW)
SolutionTosatisfytheconditions(4.1.1),kmustbenonnegative,andtosatisfythecondition
(4.1.2)wemusthave
oa
f(x)dx=
joO
1sothatk.
(Cauchydistribution柯西分布)
Example4.1.3Calculatingprobabilitiesfromtheprobabilitydensityfunction
Ifarandomvariablehastheprobabilitydensity
"xforx0
f(x)=3e
[0forx乞0
Findtheprobabilityfromthatitwilltakeonvalue
(a)between0and2;
(b)greaterthan1.
SolutionEvaluatingthenecessaryintegrals,weget
2
(a)P(0辽x乞2)=j3e'xdx=1-e"=0.9975
0
oO
(b)P(x〉1)=3e'Xdx=e"=0.0498
1
Example4.1.4
DeterminingthedistributionfunctionofX,itisknown
工3e'xforx0
f(x)=
0forx<0
SolutionPerformingthenecessaryintegrations,weget
(0forx_0
lx
"x)3e"dt=1—e"xforx0
0
P(x_1)=F
(1)=1-e‘=0.9502□
5.mean
Iftheintegral(4.1.3)doesnotconvergesabsolutely(绝对收敛),wesaythemeanofXdoesnotexist.
Definition4.1.2LetXbeacontinuousrandomvariablehavingprobabilitydensityfunction
f(x).Thenthe
mean(orexpectation)ofXisdefinedby
—E(X)二.xf(x)dx,(4.1.3)
-oO
Themeanofcontinuousrandomvariablehasthesimilarpropertiesasdiscreterandomvariable.
Ifg(X)isanintegrablefunctionofacontinuousrandomvariableX,havingdensityfunctionf(x),meanofg(X)is
oo
E(g(x))「g(x)f(x)dx
-joO
providedtheintegralconvergesabsolutely.
Example4.6.4
LetXbearandomvariablehavingCauchydistribution,theprobabilitydensityfunctionisgivenby
(a)FindE(X);
(b)Let
gw」ox:
1
0,elsewhere
FindE(g(X)).
00|x|
Solution(a)Sincetheintegral2dxdiverges(发散),E(X)doesnotexist.
丿(1+x)
1
(b)E(g(X))「g(x)f(x)dx=
_:
:
0
xln2
2dx-——
二(1x)2二
6.variance
Similarly,thevarianeeandstandarddeviationofacontinuousrandomvariableXisdefinedby
二2=D(X)=E((X7)2),(4.1.4)
WhereJ=E(X)isthemeanofX,二isreferredtoasthestandarddeviation.
Weeasilyget
□o
二2二D(X)=x2f(x)dx-」2.(4.1.5)
Example4.1.5
Determiningthemeanandvarianeeusingtheprobabilitydensityfunetion
3e"xforx0
Withrefereneetotheexample4.1.3:
f(x)-<
0forx兰0
findthemeanandvarianeeofthegivenprobabilitydensity.
SolutionPerformingthenecessaryintegrations,weget
oOoo.
1亠=xf(x)dx二x3e'xdx=_03
and
c2
:
:
11
=J(x_A)2f(x)dx=J(x__)3e:
Xdx=_
039
均匀分布
4.2UniformDistribution
Theuniformdistribution,withtheparametersaandb,hasprobabilitydensityfunetion
一forax:
b,f(x)pb-a
0
elsewhere,
whosegraphisshowninFigure4.2.1.
f(x)
1b—a
Figure4.2.1Theuniformprobabilitydensityintheinterval(a,b)
-be
Toprooff(x)dx=1.
*jtJd
Tofindthedistributionfunction.
Thedistributionfunctionoftheuniformdistributionis
- 配套讲稿:
如PPT文件的首页显示word图标,表示该PPT已包含配套word讲稿。双击word图标可打开word文档。
- 特殊限制:
部分文档作品中含有的国旗、国徽等图片,仅作为作品整体效果示例展示,禁止商用。设计者仅对作品中独创性部分享有著作权。
- 关 键 词:
- 概率论 数理统计 英文