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1、1 3 117 3.4 ? 3?SK, kIC?. X, b?,$ ?N?CX, UY = 1 2mX 2, XLX?(Y ?Q? !naK: CX?, CY = f(X)?; (X,Y )?, Z = g(X,Y )?; X = (X1, ,Xn)?, Yi= hi(X), i = 1, ,m, Y = (Y1, ,Ym)?. ?E , p()e)obf(x) 3R ?; g(x,y)3R2?R1 ?, hi3Rn?R ?. 3.4.1C? ?!, ?K, CX?FX(x)!?pk,k 1VpX(x), ?f(x)3R?Borel, CY = f(X)? !?V. , 5?y R, Y y
2、= f(X) y = X Dy, Dy= x : f(x) y. ?fN, ?Y ?. Xb?fNO, KDy= x : f(x) y = x : x f1(y), uY ? FY(y) = P(Y y) = P(X Dy) = P(X f1(y) = FX(f1(y). b?X?V, dn2.2.2!n2.2.4?, ? n3.4.1CXk?pk,k = 1,2,.(l/)VpX(x)( Y/), KY = f(X)? FY(y) = P xkDypk, l/, Z Dy pX(x)dx,Y/. ?Y = f(X)?, XJl., ?, XJ Y., V. ?e5, 5wAC?f. 1183
3、.4? 3.4.1?CXkVpX(x) = 2x,0 x 1, 0,. CY = 3X2+ 5?. ) ?y R, PDy= x R : 3x2+ 5 y, K Dy (0,1) = ,y 5, 0, r y 5 3 # ,5 y 8, (0,1),y 8. dn3.4.1, Y = 3X2+ 5? FY(y) =P(Y y) = P(3X2+ 5 y) = P(X Dy) = Z Dy pX(x)dx = Z Dy(0,1) 2xdx = 0,y 5, Z y5 3 0 2xdx,5 y 8, 1,y 8 = 0,y 5, y 5 3 ,5 y 8, 1,y 8. ?Y l!U(5,8).
4、3.4.2?CXlIO?N(0,1), CY = X2?. ) ?Y = X2?FY(y), K FY(y) =P(Y y) = P(X2 y) = P(|X| y), y 0, 0,y 0 = (y) (y),y 0, 0,y 0 = 2(y) 1,y 0, 0,y 0, 0,y 0 = (y)/y,y 0, 0,y 0 = (1/2)1/2 (1/2) y1/21ey/2,y 0, 0,y 0. 1 3 119 (t) = 1 2et 2/2 IO?V. Gamma?V?/, Y lGammaGa(1/2,1/2), =gd 1?k2(1). n3.4.2?CX?F(x)NO?Y, KCF
5、(X)l !U(0,1). y ?Y = F(X)?FY(y), K FY(y) =P(Y y) = P(F(X) y) = 0,y 0, P(X F1(y),0 y 1, 1,y 1 = 0,y 0, F(F1(y),0 y 1, 1,y 1 = 0,y 0, y,0 y 1, 1,y 1. d!U(0,1)?, =Y l!U(0,1). 5P3.4.1n3.4.2L?YCL!X, 3?k?. X|O?)15?p. ? 0, 0, = 1 y(1 + (lny)2), y 0, 0,. n3.4.4?555CCC555 ?CXl?N(,2), K?a 6= 0, CY = aX + bl?
6、N(a + b,a22). y dn3.4.3?. AO/, -a = 1/, b = /, k 3.4.1eX N(,2), KX N(0,1). 1 3 121 ?, dn3.4.3?: n3.4.5?CXlGammaGa(,), c 0, KCcX lGamma Ga ? , c ? . 3.4.2?CXl?N(0,2), KX2lGammaGa ?1 2, 1 22 ? . y d3.4.1, X/ N(0,1). d3.4.2, X2/2 2(1), = X2 2 Ga ?1 2, 1 2 ? . u, dn3.4.5 X2= 2 X2 2 Ga ?1 2, 1 22 ? . 3.4
7、.2? ?!X-?/. ?(X,Y )?, g(x,y)3R2?R1 ?Borel, ?8?CZ = g(X,Y ) ?. n3.4.6?C(X,Y )k?pij,i,j = 1,2,.(l/)V p(x,y)(Y/), KCZ = g(X,Y )? FZ(z) = P(X,Y ) Dz) = X (i,j):(xi,yj)Dz pij,l/, ZZ Dz p(x,y)dxdy,Y/. Dz= (x,y) : g(x,y) z. y 5?z R, Z z = g(X,Y ) z = (X,Y ) Dz. ud?n3.1.3, n?y. 5P3.4.2w, ?g(x,y) = x, =?CX?S
8、. = S?. 3.4.5?C(X,Y )? X Y 01 01/101/5 13/102/5 . CX?FX(x). 1223.4? ) dn3.4.6, X? FX(x) =P(X x) = P(X,Y ) (,x R) = X (i,j):(xi,yj)(,xR pij= 0,x 0, 1 10 + 1 5, 0 x 1, 1 10 + 1 5 + 3 10 + 2 5, x 1 = 0,x 0, 3 10, 0 x 1, 1,x 1. e(X,Y )?l.C, KCZ = g(X,Y )l.C, ? ?x. k?Z = g(X,Y ) kU?zk,k = 1,2,., ? XzVP(Z
9、 = zk)=. dn3.4.6?, 3.4.3e?l.C(X,Y )k?pij,i,j = 1,2,., CZ = g(X,Y )kU?zk,k = 1,2,., KZ? P(Z = zk) = X (i,j):g(xi,yj)=zk pij. 3.4.6e?l.C(X,Y )k?Xe: X Y 012 01/21/81/4 11/161/160 CZ = 2X Y ?. ) kZU?u2,1,0,1,2. d3.4.3, P(Z = 2) = P(X = 0,Y = 2) = 1 4, P(Z = 1) = P(X = 0,Y = 1) = 1 8, P(Z = 1) = P(X = 1,
10、Y = 1) = 1 16, P(Z = 2) = P(X = 1,Y = 0) = 1 16, ?d?K5 P(Z = 0) = 1 1 4 1 8 1 16 1 16 = 1 2. ?LXe: 1 3 123 Z21012 P1/41/81/21/161/16 3.4.7?CXY p, X,Y O?U/?01. CZ = max(X,Y ) ?. ) dK, XY , P(X = 0) = P(X = 1) = 1/2. Z?U?k01, P(Z = 0) =P(max(X,Y ) = 0) = P(X = 0,Y = 0) =P(X = 0) P(Y = 0) = 1 2 1 2 = 1
11、 4, P(Z = 1) =1 P(Z = 0) = 1 1 4 = 3 4. ?Z? Z01 P1/43/4 . n3.4.7lll/? ?C(X,Y )? pij= P(X = xi,Y = yj),i,j = 1,2,., KCZ = X + Y ? P(Z = zk) = X i P(X = xi,Y = zk xi) = X j P(X = zk yj,Y = yj) zk,k = 1,2,.Z?U?8. AO/, eXY p, KCZ = X + Y ? P(Z = zk) = X i P(X = xi)P(Y = zk xi) = X j P(X = zk yj)P(Y = yj
12、). y UX?, ?m? = P iX = xi, ?d Z = zk = X i Z = zk,X = xi = X i X = xi,Y = zk xi V?5=? P(Z = zk) = X i P(X = xi,Y = zk xi). e?n?. 3.4.1e,aV?C?Eda, Kdak 555. n3.4.8?555 eCXl?b(n,p), CY l?b(m,p), p, K CZ = X + Y l?b(n + m,p). 1243.4? y kCZ = X + Y ?U?k0,1,.,n + m. d(n3.4.7), P(Z = k) = X i P(X = i)P(Y
13、= k i) = X i Ci np i(1 p)niCki m pki(1 p)m(ki) =pk(1 p)n+mk X i Ci nC ki m =Ck n+mp k(1 p)n+mk, k = 0,1,.,n + m. ?A?|5, |A?K5. 5P3.4.3?p?:?. n3.4.9ttt?555 eCXltP(1), CY ltP(2), p, K CZ = X + Y ltP(1+ 2). y kCZ = X + Y ?U?k0,1,. d, P(Z = k) = X i P(X = i)P(Y = k i) = k X i=0 i 1 i! e1 ki 2 (k i)!e 2
14、=(1 + 2)k k! e(1+2) k X i=0 Ci k ? 1 1+ 2 ?i? 1 1 1+ 2 ?ki =(1 + 2)k k! e(1+2),k = 0,1,. ?|?b ? k, 1 1+ 2 ? ?K5. Y/?, k5wXY ?5?. n3.4.10?C(X,Y )?Vp(x,y), K?t R, C Z = tX + Y ?V pZ(z) = Z p(x,z tx)dx. y ?CZ = tX + Y ?FZ(z), - Dz= (x,y) : tx + y z = (x,y) : x , y z tx. u FZ(z) =P(Z z) = P(tX + Y z) =
15、P(X,Y ) Dz) = ZZ Dz p(x,y)dxdy = Z dx Z ztx p(x,y)dy = Z dx Z z p(x,u tx)du = Z z ?Z p(x,u tx)dx ? du, 1 3 125 ?1?COu = y + tx, ?g gS. uZ = tX + Y ?V pZ(z) =dFZ(z) dz = d dz ?Z z ?Z p(x,u tx)dx ? du ? = Z p(x,z tx)dx. aq/, k n3.4.11?C(X,Y )?Vp(x,y), K?t R, C Z = X + tY ?V pZ(z) = Z p(z ty,y)dy. ?t =
16、 1, nn3.4.10n3.4.11, 3.4.4X + Y ?VVV ?C(X,Y )?Vp(x,y), KCZ = X + Y ?V pZ(z) = Z p(x,z x)dx = Z p(z y,y)dy. 3.4.5X Y ?VVV ?C(X,Y )?Vp(x,y), KCZ = X Y ?V pZ(z) = Z p(x,x z)dx = Z p(z + y,y)dy. y 3n3.4.11, -t = 1=?1?. 1?n3.4.10?, d P(X x,Y y) = P(X x) P(X x,Y y) (X,Y )?Vp(x,y), udX Y = X + (Y )3.4.4=?1
17、? . 3.4.8?C(X,Y )?V p(x,y) = 2,e0 x y 1, 0,. CZ = X + Y ?VpZ(z). 1263.4? x = z 1 x = z/2 z x O 1 2 1 3-7 ?p(x,z x)? ) d3.4.4, Z = X + Y ?VpZ(z) = Z p(x,z x)dx. d? p(x,z x)? D =(x,z) : 0 x z x 1 =(x,z) : 0 z 1,0 x z/2 + (x,z) : 1 z 2,z 1 x z/2 =(x,z) : 0 z 2,max(z 1,0) x z/2. X3-7. u, Z = X + Y ?V pZ
18、(z) = Z p(x,z x)dx = Z z/2 max(z1,0) 2dx,0 z 2, 0,z 0z 2 = z 2max(z 1,0),0 z 2, 0,z 0z 2 = z,0 z 1, 2 z,1 z z) = 1 P(min(X1,X2,.,Xn) z) =1 P(X1 z,X2 z,.,Xn z) = 1 n Y i=1 P(Xi z) =1 n Y i=1 (1 P(Xi z) = 1 n Y i=1 (1 Fi(z). AO/, 1 3 129 3.4.9?CX1,X2,.,Xnp(Pi.i.d.), ? FX(x). P Y = max(X1,X2,.,Xn),Z =
19、min(X1,X2,.,Xn), KY Z?O FY(y) = FX(y)n,FZ(z) = 1 1 FX(z)n. ?/, eX1,X2,.,XnY.C, kVpX(x), KY Z?V O pY(y) = nFX(y)n1pX(y),pZ(z) = n1 FX(z)n1pX(z). 3.4.103m(0,1)/?n:, XY OLm:?I, XY ?V. ) ?n:?IOX1,.,Xn, Kd, X1,.,Xn, ?! U(0,1), X = maxX1,.,Xn, Y = minX1,.,Xn. !U(0,1)?VO p(t) = 1,0 t 1, 0,. F(t) = 0,t 0, t
20、,0 t 1, 1,t 1. ?d3.4.9, XY ?VO pX(x) = n(F(x)n1p(x) = nxn1,0 x 1, 0,. pY(y) = n(1 F(y)n1p(y) = n(1 y)n1,0 y LI2w, PY = 0; X LI2w, PY = 1. (X,Y )?9T = X + Y ?. 1303.4? ) dK?, X N(,2), Y = 1X. ?(x,y) R2, 5? X x,Y y = ,y 0, X x,Y = 0,0 y 1, X x,y 1. = ,y ,0 y 1, X x,y 1. = ,y 00 y 1x , X x,0 y , X x,y
21、1. u, (X,Y )? F(x,y) = P(X x,Y y) = 0,y 00 y 1x , FX(x) FX(),0 y , FX(x),y 1. FX(x)?N(,2)?. aq/, ?t R, T t =X + Y t = X + Y t,Y = 0 + X + Y t,Y = 1 =X t,X + X + 1 t,X = X tt + 1, X t + X t 1, t + 1, X t 1,t . ?T = X + Y ? FT(t) = P(T t) = FX(t)t + 1, FX(t) FX() + FX(t 1), t 0,y 0; 0,/. C u = x + y,
22、v = x/y. ?x 0,y 0, _C x = uv 1 + v , y = u 1 + v . du 0, v 0. NO 1? J = ? ? ? ? ? ? ? x u x v y u y v ? ? ? ? ? ? ? = ? ? ? ? ? ? ? ? v 1 + v u (1 + v)2 1 1 + v u (1 + v)2 ? ? ? ? ? ? ? ? = u (1 + v)2 . u, dn3.4.15, (U,V )?V p(u,v) = eu u (1 + v)2 ,u 0,v 0; 0,/. 1 3 133 ?/, U?V p U(u) = Z p(u,v)dv = Z 0 eu u (1 + v)2 dv,u 0; 0,u 0. = ueu,u 0; 0,u 0. 5P3.4.9l3.4.12?u, U = g(X,Y )?, ? ?, kO#?CV = h(X,Y )(5? -V = XY ), |n3.4.15k(U,V ) ?2U = X +Y ?S. OOOCCC, 4.