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V pLc ξ, X ‹ G z£ p( x)D, ˝ ˇx41,›x2,(Xx1<x2) ¨ xP2(x1<ξ≤x2)=F(x2)−F(x1)= p(y)dy 2˜.4 ¯x1E› 4§ p , X \·1 T) ξ ):ª ( x1,x2] ,X ¨VD) [Q1 b( x1, x 2] K¨ + y˘=4p (x) 6 ,X ˘E 5¨ 6 ß,BX2 M6>¯</ ˆ ¨H y˘=4p (x) „ x˜EH „,¯ XM6/1. ˆ + 2˜.4 ¯ªE‹ ˆ „E¨A† 4 ` _Lc ‹G£ L“˚ ¨)G& A ¨˝¨, Xˇ ªV)[ ,Xx,P{ξ=x}=0, b P{x1≤ξ≤x2)=P(x1<ξ≤x2)+P(ξ=x1) x=P2(x1<ξ≤x2)= p(y)dy x1 V p(px) ⁄ 8 ¨ Y, X ¨ D + 2˜.4 ! ¯ª¤ EˆW - L¨„ßc ‹G£: E› 8 ¨ Y,X V )E¨[› 3ªE WpG( x-)ß, K z ,X ß B ¨ „/˜ W V)[ z D!¨8 +Œ¨ 2˜.3 -¯ „ ¨˝p(x),XE†4` & dF(x)=F′(x)=p(x). dx M6 4¡ · ? ,X. E†4` _Lc ‹G£1 ˚ 5 <%1 ˆ 6 [a K,b ]¨ Y i,&X K´ NE¨l›G 1 ˆ6 ,X J Æ A¨ i,X& (ξ): [a ,b] Y!£ & ,X ˆ5¨6 ßA,¨ i ,X[a&,b ] :Y ˇ $ K¨,X V)[ A„ $ [aK,b¨] Y ,X K ’ 5¨! 5 B · $G K¨., XKS z G kξ, X D 0x<a Fx−a(x)= a≤x<b b−a
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A¨ ξ~U[a,b]. [ _1] @ E"QE:100 › !J£sL h EZ " ¨Q,E v:1 E0 EJ s Y ˚ˇ E "QE:0› 1 "¨ ˆ,6 v, X E:3 Æ CJYsE, .X V)[? •A ξ></ , v ¨E :ξ~ U˚[0,K5]¨, ξ,X z D 1 0<x<10p(x)= 10 0 J “ 313 P{0≤ξ≤3}= dx= 01010[ _2] A ξ [ 0 5¨] Æ ¢ "¨ / 4 x2+4ξx+ξ+2=0 r ,X V.) [? • + Nlξ,AX V)[ z D 1 =0≤x≤5p(x) 5 ¨ 0 J “ ˝ ‘ ı ¨ ł/ ª∆=[4ξ2]−4×4[ξ+2]=16(ξ−2)(ξ+1)≥0 G ξ’≥2 ˚ ¨ / ¨ ξr≤ − 1 Æ N l7¨ ƪ. „ / r ,X V)[ 513P{P{ξ≥2}= dx= . 2552 !˚7 −µ2(x)8„1− µ,σ(σ>0) ł ¨(D)=e2σ2px−∞<x<+∞ /˜ !7 2piσ z !¨ 7D n,X . A/ ˜ ξ ~ N!(µ7,σ 2) !7 V)[A ¨ GaGus¡s ? U-,ŁX0 J A ˆ ´ ) A ˚ ¨ * W 9 + Aˆ ´ „ o:+ G0au ss 3. 4/ £˜P ‘ ><A¨ rL K´N l ¨ V #, XG £‹ ¨AG⁄ˆ£ ´)`5 A' ¨ ⁄4 ‰ H+ $,XD P‹F .Eˆfl „ A!x9 , X )Æ A ¢ ¨!- 7Ł 0 J > < ‹G£8„ « (ˆß G0£ Lˆ , cX ª ·,2X , XF¨ Ew E¡ ¨› ‹G£ 8 ¨ !7 ‹G£E› & E flD ) !49.‡ -AŁu0J ¢!7 z D,X ¨p (x )6 Gˆ x b=„µ,& ˝˛ / ¨˜ x=µ E ’¨U ßµ ˛ n σ¨˚,X ¨p (ªx),X 6 ˆ ( `¨Œ σ,X C¨^p( x)ß,X 6 ˆ G
. + !M6 G b V)[ z ¨ VD ,p(pXx)A | A ˘4£-„F’µ& L E¥ ˆ PŒ ‹ ¨ Lc µ‹&G £L E ¥ “ ,X V)[C^. ß_ r ¨˝ ˇ NƵ ¢2(,σ),XLc ξ ‹ G£ P(µ−σ≤ξ≤µ+σ)≈ P(µ−2σ≤ξ≤µ+2σ)≈ P(µ−3σ≤ξ≤µ+3σ)≈ E›A ¨ „¨ µ L ¨c ξ ‹,GX£4– ˝ σ Æ,CXY EV )[+2/39 P ¨Æ‹C Yb2Eσ ,X V)[95 % „ ¨5 CY3σE, X V)[ 03 . · E› V ¨) [ r \L ªK ´ ¨ N l A x W Æ E¨¥›* , X 4‡Au | 4|£ * ˜s Œ) , X ªr LV )[ .¯ _ ˚ s) rL | ª sV)) [ _ ˚ ıA' P¥‘* . , X· " Æ ˆ6 ÆE h "… E¨ª› G · " Æ ˆ 6 J \¥ * ˝ ıA ' P9‘A ¨ ¨,·X V pA'P‘ ı D AZ¨„ _ ˚ ’’ \. ˆ6 ¥* ,X"… ª !7 3σ‹8G £ :¨ Œ,X0 .00V3 ) ¨[ ª V ) ¨[ „_ r˚L 4‡Au 4£3σ 8* ¨ Y!7 z ˘4 ,XM6/ˆ 9 E¨•› H r L! 74 ‡ A˘u4 ,XM6/ˆ4£ * 3 σ, Xs . 8„ξ~Nµσ2(,) +¨ ! 8V G b D,XA|A -„−2(xµ)1x−P2(x1ξx)=F(x)−F(x)=e2σ2<≤221 dx 2piσx1) 4 ,X :!¨ 7ˆ 6 K,´X/ˆ VD :) k d, X ˘4£ ‘ ,X o D / ˆ ˜ _ ,VX ^ > "/'ˆ D* ¯ ~E4›{G D ’)’ ƨ*1E ¨k! M6 ˘4£ !¨ 7˛ V)[A ¨ *Z F,SX ! ! £ ı .¨E›/¡4 ,XAu1k A ˙ !ˆ7 „ 4 Œ ><. „ o! ¨ 7„ * µ ,σ (σ > 0 ) 4¨D› n Æ ,Xµ σ˝ ¨ Æ , X ¨’!’7 Æ ˆ 6 µ˝ σ F, X4Œ ˝ h,X!7 . _ r > <¨ 4µŒ= 0 ¨σZ=1,X!7 >< „ ¨No(0 ,1„) * /˜ ! ¨7J z φ( x)D *9 >< / ¨ Φ (xD) ¨ G 2yΦx1x−(x)= φ(y)dy=e2dy −∞ 2pi−∞ :,XL )N4(0›,1) ˛ Z Φ,(xX)>< F¨w ?NU „2(µ,σ) E › r ?UE E ‹ 6 . 6 ?• K´Nl2(y−µ)A 1x−ξ N2(µ,σ2),XLc ¨ ‹ GP( £ξ≤x)=e2σ dy 2piσ−∞
‚ξ−µη= ¨ η 3 Lc ¨J Ł‹ G £σy−µ2()ξ−µ1σx+µ−P(η≤x)=P(≤x)=P(ξ≤σx+µ)=2σ2 σ2piσ edy−∞ ˝/ˆ ‹ G0£ ¨•‚ 6y−µu= ¨G kσu21x−P(η≤x)=e2du=Φ(x) 2pi −∞+ !8 ˆη¨ ? NÆ(0 ,1¢) ,X ¨!b7 ? FU‹(x G)„£=P(ξ≤x) ¨ ?UΦ (y„) G ˆ¨J x−µy= ¨· E › ˚ σξ−µx−µx−µx−µF(x)=P(ξ≤x)=P(≤)=P(η≤)=Φ(). σσσσ A ξ~N(0,1) A¨u1P{kξ<),P{ξ<−},P{ξ<} ,¨ y „ ! 7 >< kP{ξ<}=Φ()= P{ξ<−}=Φ(−)=1−Φ()=1−= P{X<}=P{−<X<}=Φ()−Φ(−)=2Φ()−1=2× =. [ _3] A Lc ξ ~‹NGµ£σ2(,)(σ>0) ¨Ł ‘ ıy 2 +4/y +ξ=0 · r ,X V)[ 1 "¨ µ,X. 2?• y2+4y+ξ=0 · r 11 6− 4¸ξ <b0 ¨Gξ>4. 1Φ 4−µP ={ξ>4}=1− 2
σ Φ 4−µ 1 =
σ 24−µ1= σ2 µ=4 [ _4] A ⁄ <* { ,c˜Xm= ¯ Æ ,¢X µK =S1 0 ,σ=,X!7 ¨
„8N? nKS1 ± 0 .1¨2 Y "¨ = ,X. ı )[?• P{−≤ξ≤+} Φ+− =()Φ−−−() =2Φ(2)−1 = ı 1−)0[.95 4 5= [ _5] ⁄ 3 3fl0 6 , ¨ 5 A' 4‰ ¢P‹ ¨ E "10 00 q y ı5¨Æ ) “ y Æ5 ,X5ξ ~AN'(µ ,σ42)‰ ¨˘-90„ „ 3 6 6¨0 „ 1 15 K¨´> ) “5 " D ?• + N lA 90−µ36P{ξ>90}=1−P{ξ≤90}=1−Φ()== σ100060−115P{ξ<60}=Φµ()== σ1000 ¡ „!7 >< k 90−µ =σ 60−µ = σ?• k µ=72 σ=10 ξ~N12(72,0). A > )* 5 x0 " ¨ > D) * 10)00[− 33 0= 100072P{ξx}Φx0−<0=()= 10x0−72= 10x0= G> )* 5 76 ". D 3. D A Lc ξ, X‹ G £ z D λ−λxe,x>0p(x)= λ>0 0,x≤0
/ξ ˜Æ ¢ λ ,DX D. D 3 V)[4‡Au ƨ , X 2*O G ¡{? rUC 5¨ SŁ" fl ,X h* /¥ :-Ł0J U JG¡?U5 (M ! ^·,AX 0 ˜ *·_ )G K 3b WK K ,X ßB .¯ A Lc ξ Æ‹ G¢£ λ ,DX D ¨ ˝ ˇs >ª0,,tX>0, P{ξ≥s+t} P{ξ≥s+t|ξ≥s}= P{ξ≥s}1−F(s+t) = 1−F(s)−λ(s+t)e ==−λte=P{ξ≥t}. −λte V pξ) ^ ?• ¨ Q ª >< ¨ · A ⁄/¡ { > S*s ¨Z ? UKES‹ " u! ˚K¨ 3 W¨#6 a S* t , X! V )˚[K ¨ ˚ { t, X6 V )S[ * ¨ G E › /˚¡K ¨{ " E H ,› & 3A¨ „ D .0 1 u V ¨Q! 8 \ 5 LK,X E‹ ªG * E›/¡ bAu1k. ,X 0 { S* Q,X ı _[ _6] S*t ª Z˚,X&` ∆#t +Y L ,Xλ∆ tV+o)(∆[t) A¨ A„&` #+ L ,X S* Q E† 4"¨` A_„L&c` # +‹ GL£ ,X . S* Q,X ?• * ξ></A „&` #+ L ,?¨XU "S *, FX( x ) =QP{ξ≤x}.+ Nl ª kP{t<ξ≤t+∆t}P{t<ξ≤t+∆t|ξ≥t}==λ∆t+o(∆t) P{ξ≥t} GF(t+∆t)−F(t) =λ∆t+o(∆t),t≥0 1−F(t) J 1− F(t)=F(t)/˜ ξ, X* ,. ‘ <D2O ¨˝ t > 0 ¨’s=t−∆t>0 ˚ ¨ F(t)−F(t−∆t)=P{t−∆t<ξ≤t|ξ>t−∆t} 1−F(t−∆t) =P{s<ξ≤s+∆t|ξ>s]=λ∆t+o(∆t). b ˝ M6 ł ∆t ¨ aL ∆8t‚→ „0 ¨k F′(t) =λ,t≥0. F(t)
ł0ˆ/ˆ F (t )k=C−λte C¨ . aD F(*0) =C=1,0 G Fˆ =k−λt(t)e ¨b b Fλt(t)=1−−e,t≥0." — D k zλλx −e,x≥0 pD(x)= ¨G . D 0,x<0 c L c ‹ GD£,X V)[A ,X r ¨L˚ K ´ NFl " Lc ‹G.£ _ VD , X4 ‡ A ¨u (,=X)K ´ N l ˘-„!Ł ’ X ,$X, X E?¨ U z" $,1XmX 2|,6X ¨ !¤ V ˘-„ „ ˚,X Y X2 Æ ¢,X ?¨ U " „ ˚,X M6. / ˆ„,EX› 8 V 1 1 A|A Au1kLc ‹G£ D ,X *. "' 8„X / 7 _ L ¨cV ) [ ‹ GP{ £ξ = x }=p,i=1,2, .Y=g(X) X,X ii D )¨ ?UY", X . 1˜ 8¯„y=g(x), f Æ ¨, Y ,1X P{ Y = y}=p,i=1,2,3, . iiii 2˜ 8¯„y=g(x), Æf Æ , ¨1 Y ,X P{ Y = y}= P{X=x}. iiijg(x)=yji[ _1] A X,X º X −2−1013 P " Y=X2,X . ? • Y,X “ 0 1¨ 4¨ 9¨ ¨Ł P{Y=0}=P{X=0}=; P{Y=1}=P{X=1}=+=; P{Y=4}=P{X=−2}=; P{Y=9}=P{X=3}= b Y ,X
Y 0 1 4 9 P [ _2] ˘-„Lc X , X‹ G £ P{ Xº= k} =ak(k=1,2, ),.B n a ¨DJ" piY=sin(X),X . º2∞∞a ? •+ 1= p= akk= ¨G k1a=. k=1k=11−a2 + Nl ª ˆ k ><X 1 2 3 4 5 6 7 8 … V345678 ) [ 1/2 1/22 1/2 1/2 1/2 1/2 1/2 1/2 … Y=sinpiX2 1 0 -1 0 1 0 -1 0 … pi Y=sin(X) : -1ˆ,06,1, 211112 P{Y=−1}=+++ == 23272118×(1−1)151611111 P{Y=0}=+++ == 2224264×−1(1)34218 P{Y=1}=1−−=. 15315 Y,X º Y -1 0 1 218 P 15 3 15 MA6|A E†4` _Lc ¨‹ G,£ D , KX ’ _ $ >_ A @Lc X ‹ÆG £¢ U − pi pi(,) Y¨=tanX A¨'"Y ,X D ‘ z 22 D. ?• - ¨„˝ ˇ )x r¨ D
piFY(x)=P{Y≤x}=P{tanX≤x}P{Xarctan,∈pi=≤xX(,)} 22arctanx111 = pidt=arctanx+ −2pipi2E› GY = tanX,X . + ! 8D"Y , kX z D d1 pY(x)=FY(x)=,x∈R dxpi+2(1x)E› Ca uchy ,X(.M !^ 6 8 ¨ z D 1λ2 p(x)= 2˜.5 ¯piλ2+x−µ2(),XE†4` _ Ca uc hy / ˜ . _3 ,YX Æ ¢,λX=1 ,µ =0,XCauchy . Ca uchy V)[A ,X /¡. K •>< ß,X 3_,X | —¨ 5 ˚pipix∈(−,)CK ZG¡? U ¨ ·0 * t ∈ ’pipi(−,) ˚ ¨ D2222u=tant ø ¨¢5 W , ‡ ,X ¡t= a rc tanDu !¨8 ˚ (tanX<x)=(X<arctanx),X G2ˇ. Jª\ ˝0E ›/ ¡ ¨ 4 › ˛ . V n) n)2 .2 A Lc X ‹ÆG £¢E†4`FX (x_) ,¨ h,X zp X (x ),aD< x <b ¨5 u =Dg(t) (Ka,b)¨ ,X ø )A , X ¨WE,†X4 `¡ h u = DgD−1()(u) ⁄ K(α¨,β) ,X ˆ — Ł¨ DJ — dh( u)D=g−1′(u) (αK,β)¨ E† 4` ¨ duY=g(X) E†4` _L c ¨ J ‹zG £ D pY(x)=pX(h(x))h′(x),x∈(α,β). 2˜.6 ¯A _ r ’¨g ø ¨ ˚ FYx)=P{Y≤x}=P{g(X≤=≤−1()x}P{Xg(x)} =P{X≤h(x)}=FX(h(x)), ’g ø L ¨! ˚ F()={≤}=P{gX≤=−1YxPYx()x}P{X≥g(x)} =P{X≥h(x)}=1−FX(h(x)).
„ ¨ E˜ ł/ ¡¨ d pY(x)=FX(h(x))=pX(h(x))h′(x)=pX(h(x))h′(x),x∈(α,β). dx ‘d pY(x)={1−FX(h(x))}=−pX(h(x))h′(x)=pX(h(x))h′(x),x∈(α,β). dx[ _3] A Lc X ,‹XG £ z D −xe,x≥0 p(x)= 0,x<0" Lc Y =‹eG3X£,X z. D?• ’y≥1 ˚4lnxlnx1− pY(x)=pX()()3 ′=x 333 41− x3 =≥ p,x1Y(x) 3 0, J “ F ,XA ¨y = gK(x ) ’ÆK ´ Nøl )!¨A8 ,˚X (−ˆ∞, +^∞) 8„ F !¨ª¤ ∆kVK(k¨=1,2, ) ¨S yk=g(x) ∆k(k=1,2, ) ø ¨)JA ¡ Dg− 1k( y), Y,X z D pg−1−1X(k(y))(gk(y))′, pY(x)= k 2˜.7 ¯ 0,8y„ S ¡ D Æ , [ _4] RX ¨ s& ,X "¨< A „ &ˇ “X , X$& z. D? • A Lc M (x&,y ) ˝ h θ? ƒ+¨ Nlθ ª (−pi,pi) Æ ¢ ¨ J z D 1 ¨piθpi, =−<<pθ(θ) 2pi 0, J, “+ X=Rcosθ ¨ (−pi,0) X=Rcosθ )A E ¨ ¡r θD= −a xrccos R(0,pi) X=Rcosθ )A E¨ ¡ £ θD= xarccos. R
1x1x b (−arccos)′+(arccos)′ ¨−R≤x≤R , pX(x)= 2piR2piR 0, J, “ 1 „ ¨−R≤x≤R, pX(x)= piR2−2x 0, J. “[ _5] A X Æ ¢ !"¨7 Y = X 2 , X . ?:• ’Y =X2 Æ “B ¨ · !8y ≤’0 ˚Y¨,X F( y )= 0D ’y>0 ˚ F{2(y)=PX≤y}=P{−y≤X<0}+P{0<X≤y} x2x201−y1− =e2 dx+e2dx −y2pi 02pi" — ˆY =kX2,X z Dy1− p(y)=e2,y>0 2piyE› n 7 + χ 2z , X z Dny1−1− pn(y)=y2e2,y>0n 2˜.8 ¯n22Γ()2 ’n=1 ˚,X(. M„ _ A ¨ V X p1,X2, Xn, f( 0 Æ ¢ ¨ !7 nY= X2k Æ ¢ z 2˜ .8 D ¯ª+4 › χ˛2, X . k=1
