已知二维随机变量(X,Y)的联合分布函数为F(x,y),则P(X>a,Y>b)=__ 设二维连续型随机变量(X,Y)的联合分布函数为F(x,y)=...
\u7528(X,Y)\u7684\u8054\u5408\u5206\u5e03\u51fd\u6570F(x,y)\u8868\u793ap(a<X\u2264b,Y\u2264c)=p(a<X\u2264b,Y\u2264c)=F\uff08b,c)-F(a,c)\u3002\u5728\u5b9e\u9645\u95ee\u9898\u4e2d\uff0c\u5e38\u5e38\u8981\u7814\u7a76\u4e00\u4e2a\u968f\u673a\u53d8\u91cf\u03be\u53d6\u503c\u5c0f\u4e8e\u67d0\u4e00\u6570\u503cx\u7684\u6982\u7387\uff0c\u8fd9\u6982\u7387\u662fx\u7684\u51fd\u6570\uff0c\u79f0\u8fd9\u79cd\u51fd\u6570\u4e3a\u968f\u673a\u53d8\u91cf\u03be\u7684\u5206\u5e03\u51fd\u6570\uff0c\u7b80\u79f0\u5206\u5e03\u51fd\u6570\uff0c\u8bb0\u4f5cF(x)\uff0c\u5373F(x)=P(\u03be<x) (-\u221e<x<+\u221e)\uff0c\u7531\u5b83\u5e76\u53ef\u4ee5\u51b3\u5b9a\u968f\u673a\u53d8\u91cf\u843d\u5165\u4efb\u4f55\u8303\u56f4\u5185\u7684\u6982\u7387\u3002
\u6269\u5c55\u8d44\u6599\uff1a
\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\u7684\u5206\u5e03\u5f8b\u548c\u5b83\u7684\u5206\u5e03\u51fd\u6570\u662f\u76f8\u4e92\u552f\u4e00\u51b3\u5b9a\u7684\u3002\u5b83\u4eec\u7686\u53ef\u4ee5\u7528\u6765\u63cf\u8ff0\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\u7684\u7edf\u8ba1\u89c4\u5f8b\u6027\uff0c\u4f46\u5206\u5e03\u5f8b\u6bd4\u5206\u5e03\u51fd\u6570\u66f4\u76f4\u89c2\u7b80\u660e\uff0c\u5904\u7406\u66f4\u65b9\u4fbf\u3002\u56e0\u6b64\uff0c\u4e00\u822c\u662f\u7528\u5206\u5e03\u5f8b(\u6982\u7387\u51fd\u6570)\u800c\u4e0d\u662f\u5206\u5e03\u51fd\u6570\u6765\u63cf\u8ff0\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\u3002
\u6982\u7387\u5206\u5e03\u51fd\u6570\u662f\u968f\u673a\u53d8\u91cf\u7279\u6027\u7684\u8868\u5f81\uff0c\u5b83\u51b3\u5b9a\u4e86\u968f\u673a\u53d8\u91cf\u53d6\u503c\u7684\u5206\u5e03\u89c4\u5f8b\uff0c\u53ea\u8981\u5df2\u77e5\u4e86\u6982\u7387\u5206\u5e03\u51fd\u6570\uff0c\u5c31\u53ef\u4ee5\u7b97\u51fa\u968f\u673a\u53d8\u91cf\u843d\u4e8e\u67d0\u5904\u7684\u6982\u7387\u3002
\u5229\u7528\u6982\u7387\u5206\u5e03\u51fd\u6570\u7279\u6027
F(\u6b63\u65e0\u7a77,\u6b63\u65e0\u7a77)=1,
F(\u8d1f\u65e0\u7a77,\u8d1f\u65e0\u7a77)=0,
\u5e26\u5165\u5c31\u662f
A(B+\u03c0/2)(C+\u03c0/2)=1
A(B-\u03c0/2)(C-\u03c0/2)=0
\u5c55\u5f00\u540e,\u4e24\u5f0f\u76f8\u52a0\uff1a
ABC=1/2-(\u03c0^2)/4
答案为:F(a,b)+1-[F1(a)+F2(b)]
由于F(a,b)=P{X≤a,Y≤b},F1(a)=P{X≤a,Y<+∞},F2(b)=P{X<+∞,Y≤b},
而:
P{X>a,Y>b}=P{X<+∞,Y<+∞}-P{X≤a,Y<+∞}-P{X<+∞,Y≤b}+P{X≤a,Y≤b}
∴P{X>a,Y>b}=1-F1(a)-F2(b)+F(a,b)=F(a,b)+1-[F1(a)+F2(b)]
扩展资料:
联合概率分布的几何意义与定义
设(X,Y)是二维随机变量,对于任意实数x,y,二元函数:
F(x,y) = P{(X<=x) 交 (Y<=y)} => P(X<=x, Y<=y)
称为:二维随机变量(X,Y)的分布函数,或称为随机变量X和Y的联合分布函数。
随机变量X和Y的联合分布函数是设(X,Y)是二维随机变量,对于任意实数x,y,二元函数:F(x,y) = P{(X<=x) 交 (Y<=y)} => P(X<=x, Y<=y)称为二维随机变量(X,Y)的分布函数。
如果将二维随机变量(X,Y)看成是平面上随机点的坐标,那么分布函数F(x,y)在(x,y)处的函数值就是随机点(X,Y)落在以点(x,y)为顶点而位于该点左下方的无穷矩形域内的概率。
1-Fx(a)-Fy(b)+F(a,b)
1-F(a,+∞)-F(b,+∞)+F(a,b)
过程
绛旓細6.浜岀淮杩炵画鍨嬮殢鏈哄彉閲,宸茬煡浜岀淮闅忔満鍙橀噺(X,Y)鐨鑱斿悎姒傜巼瀵嗗害涓篺(x,y),鏈汉鍩虹杈冨樊,鎵浠ヨ璇︾粏浣滅瓟,婊℃剰涓瀹氬ソ璇!鏄鍏!!!璋㈣阿!!!... 6.浜岀淮杩炵画鍨嬮殢鏈哄彉閲,宸茬煡浜岀淮闅忔満鍙橀噺(X,Y)鐨勮仈鍚堟鐜囧瘑搴︿负f(x,y),鏈汉鍩虹杈冨樊,鎵浠ヨ璇︾粏浣滅瓟,婊℃剰涓瀹氬ソ璇!鏄鍏!!!璋㈣阿!!! 灞曞紑 鎴戞潵绛 1...
绛旓細棰樹笉闅撅紝璁$畻澶鏉傦紝缁欎綘涓濊矾 (1) 1=鈭(0,+鈭)dx鈭(x,+鈭)ke^(-y)dy=k (2)P(X+Y<=1)=鈭(0,1/2)dx鈭(x,1-x)e^(-y)dy (3)fX(x)=鈭(x,+鈭)e^(-y)dy=e^(-x) (x>0)(4)fY(y)=鈭(0,y)e^(-y)dx=ye^(-y) (y>0)f(x|y)=f(x,y)/fY(...
绛旓細绗簩闂紝鍦ㄧ涓闂殑鍩虹涓婃眰寰椾簡瀵嗗害鍑芥暟锛屽垎鍒叧浜巟鍜寉绉垎鍗冲緱鍒皔鍜寈鐨勮竟缂樺瘑搴 绗笁闂紝鏍规嵁cov(x,y)鐨瀹氫箟姹傦紝鍗砪ov(x,y)=E(x-E(x))(y-E(y))鍗冲彲銆傞渶瑕佸湪瑙i涓嚜宸辨荤粨鏌愪簺鎶宸э紝璁颁綇涓鐐癸細鍦ㄦ鐜囩粺璁′腑锛屽彧瑕佺煡閬撲簡涓涓闅忔満鍙橀噺鐨勫瘑搴﹀嚱鏁版垨鍒嗗竷鍑芥暟锛屽氨鈥滀竴鍒囬兘鐭ラ亾鈥濅簡銆
绛旓細璁浜岀淮闅忔満鍙橀噺锛坸,y锛夌殑鑱斿悎鍒嗗竷鍑芥暟涓 F锛坸,y锛=a(b+arctan(x/2)缁欎綘涓濊矾鍚,杩欎釜涓嶅ソ鎵 1) 鐢盕(鏃犵┓,鏃犵┓)=1,F(璐熸棤绌,璐熸棤绌)
绛旓細鏍规嵁瑙勮寖鎬у拰褰掍竴鎬 k=1/3锛沺锛坸y锛=1-p锛坸锛-p锛坹锛=1-1/2-1/3 =1/6 E(xy锛=E锛坸锛*E锛坹锛=1/6 瀹氫箟 涓鑸紝璁綞鏄竴涓殢鏈鸿瘯楠岋紝瀹冪殑鏍锋湰绌洪棿鏄疭={e}锛岃X=X锛坋锛夊拰Y=Y(e)S鏄畾涔夊湪S涓婄殑闅忔満鍙橀噺锛岀敱瀹冧滑鏋勬垚鐨勪竴涓悜閲锛圶锛孻锛锛屽彨鍋浜岀淮闅忔満鍙橀噺鎴栦簩缁撮殢鏈哄悜閲忥紱鏈...
绛旓細瀵逛簬浠绘剰瀹炴暟x,y锛屼簩鍏冨嚱鏁帮細F(x,y) = P{(X<=x) 浜 (Y<=y)} => P(X<=x, Y<=y)锛涢殢鏈哄彉閲廥鍜孻鐨勮仈鍚堝垎甯冨嚱鏁版槸璁(X,Y)鏄簩缁撮殢鏈哄彉閲忥紝瀵逛簬浠绘剰瀹炴暟x,y锛屼簩鍏冨嚱鏁帮細F(x,y) = P{(X<=x) 浜 (Y<=y)} => P(X<=x, Y<=y)绉颁负浜岀淮闅忔満鍙橀噺(X,Y)鐨鍒嗗竷鍑芥暟銆
绛旓細銆愮瓟妗堛戯細F(-鈭,-鈭)=A(B-蟺/2)(C-蟺/2)=0 F(-鈭,+鈭)=A(B-蟺/2)(C+蟺/2)=0 F(+鈭,-鈭)=A(B+蟺/2)(C-蟺/2)=0 F(+鈭,+鈭)=A(B+蟺/2)(C+蟺/2)=1 瑙e緱锛欰=1/蟺^2,B=蟺/2,C=蟺/2 f(x,y)=dF(x,y)/dxdy=1/[蟺^2 (1+x^2)(1+y^2)]...
绛旓細D(y)=19/320銆傜被鏉′欢姒傜巼瀵嗗害鏄紝鍋囧畾x鏄竴涓繛缁闅忔満鍙橀噺锛鍏跺垎甯冨彇鍐充簬绫诲埆鐘舵侊紝琛ㄧず鎴恜(x|蠅)鐨勫舰寮忥紝杩欏氨鏄滅被鏉′欢姒傜巼瀵嗗害鈥濆嚱鏁帮紝鍗崇被鍒姸鎬佷负蠅鏃剁殑x鐨勬鐜囧瘑搴﹀嚱鏁帮紙鏈夋椂涔熺О涓虹姸鎬佹潯浠舵鐜囧瘑搴︼級銆傛墍浠ョ被鏉′欢姒傜巼瀵嗗害鏄暟瀛︿腑杈冨父鐢ㄧ殑鏁板鏂规硶銆傝礉鍙舵柉鍏紡锛1銆佽礉鍙舵柉鍒嗙被鍣ㄤ緷鎹被鏉′欢...
绛旓細P(X+Y>=1)锛濃埆鈭紙x+y鈮1锛塮(x锛寉)dxdy 锛濃埆鈭紙0<x<1锛0<y<2,x+y鈮1锛夛紙x锛2锛xy锛3锛塪xdy 锛濃埆鈭紙0<x<1锛1-x鈮<2锛夛紙x锛2锛媥y锛3锛塪xdy 锛濃埆锛0锛1锛塪x鈭紙1锛峹锛2锛夛紙x锛2锛媥y锛3锛塪y 锛濃埆锛0锛1锛墄锛2路y锛媥锛6路y锛2锝滐紙1锛峹锛2锛塪x 锛濃埆锛0...
绛旓細宸茬煡浜岀淮杩炵画闅忔満鍙橀噺X锛孻鑱斿悎瀵嗗害鍑芥暟 f(X,Y),鏍规嵁浜岀淮杩炵画闅忔満鍙橀噺鐨瀹氫箟锛氣埆鈭玣锛圶锛孻锛dxdy=1銆傝繖閲岀Н鍒嗕笂涓嬮檺涓簒锛寉鐨勫彇鍊艰寖鍥达紝鍙眰瑙e嚭甯告暟c銆傝竟缂樺瘑搴﹀嚱鏁帮細fX(x)=鈭玣锛坸,y锛dy,绉垎涓婁笅闄愪负y鐨勫彇鍊艰寖鍥 fY(y)=鈭 f锛圶,Y锛dx,绉垎涓婁笅闄愪负x鐨勫彇鍊艰寖鍥 ...