已知，二位随机变量（X,Y）相互独立，即有F（x,y）=F(x)F(y)， 大学概率论题。 已知随机变量(X,Y)服从二维正态分布，其联...
\u5df2\u77e5\u4e8c\u7ef4\u968f\u673a\u53d8\u91cf(X,Y)\u7684\u8054\u5408\u5206\u5e03\u51fd\u6570\u4e3aF(x,y),\u5219P\uff08X>a,Y>b)=\uff3f\uff3f\u7b54\u6848\u4e3a\uff1aF\uff08a\uff0cb\uff09+1-[F1\uff08a\uff09+F2\uff08b\uff09]
\u7531\u4e8eF\uff08a\uff0cb\uff09=P{X\u2264a\uff0cY\u2264b}\uff0cF1\uff08a\uff09=P{X\u2264a\uff0cY\uff1c+\u221e}\uff0cF2\uff08b\uff09=P{X\uff1c+\u221e\uff0cY\u2264b}\uff0c
\u800c\uff1a
P{X\uff1ea\uff0cY\uff1eb}=P{X\uff1c+\u221e\uff0cY\uff1c+\u221e}-P{X\u2264a\uff0cY\uff1c+\u221e}-P{X\uff1c+\u221e\uff0cY\u2264b}+P{X\u2264a\uff0cY\u2264b}
\u2234P{X\uff1ea\uff0cY\uff1eb}=1-F1\uff08a\uff09-F2\uff08b\uff09+F\uff08a\uff0cb\uff09=F\uff08a\uff0cb\uff09+1-[F1\uff08a\uff09+F2\uff08b\uff09]
\u6269\u5c55\u8d44\u6599\uff1a
\u8054\u5408\u6982\u7387\u5206\u5e03\u7684\u51e0\u4f55\u610f\u4e49\u4e0e\u5b9a\u4e49
\u8bbe(X,Y)\u662f\u4e8c\u7ef4\u968f\u673a\u53d8\u91cf\uff0c\u5bf9\u4e8e\u4efb\u610f\u5b9e\u6570x,y\uff0c\u4e8c\u5143\u51fd\u6570\uff1a
F(x,y) = P{(X P(X<=x, Y<=y)
\u79f0\u4e3a\uff1a\u4e8c\u7ef4\u968f\u673a\u53d8\u91cf(X,Y)\u7684\u5206\u5e03\u51fd\u6570\uff0c\u6216\u79f0\u4e3a\u968f\u673a\u53d8\u91cfX\u548cY\u7684\u8054\u5408\u5206\u5e03\u51fd\u6570\u3002
\u968f\u673a\u53d8\u91cfX\u548cY\u7684\u8054\u5408\u5206\u5e03\u51fd\u6570\u662f\u8bbe(X,Y)\u662f\u4e8c\u7ef4\u968f\u673a\u53d8\u91cf\uff0c\u5bf9\u4e8e\u4efb\u610f\u5b9e\u6570x,y\uff0c\u4e8c\u5143\u51fd\u6570\uff1aF(x,y) = P{(X P(X<=x, Y<=y)\u79f0\u4e3a\u4e8c\u7ef4\u968f\u673a\u53d8\u91cf(X,Y)\u7684\u5206\u5e03\u51fd\u6570\u3002
\u5982\u679c\u5c06\u4e8c\u7ef4\u968f\u673a\u53d8\u91cf(X,Y)\u770b\u6210\u662f\u5e73\u9762\u4e0a\u968f\u673a\u70b9\u7684\u5750\u6807\uff0c\u90a3\u4e48\u5206\u5e03\u51fd\u6570F(x,y)\u5728(x,y)\u5904\u7684\u51fd\u6570\u503c\u5c31\u662f\u968f\u673a\u70b9(X,Y)\u843d\u5728\u4ee5\u70b9(x,y)\u4e3a\u9876\u70b9\u800c\u4f4d\u4e8e\u8be5\u70b9\u5de6\u4e0b\u65b9\u7684\u65e0\u7a77\u77e9\u5f62\u57df\u5185\u7684\u6982\u7387\u3002
\u5148\u6839\u636exy\u7684\u4e8c\u7ef4\u5206\u5e03\u7684\u6807\u51c6\u5f62\u5f0f\u5206\u522b\u6c42x\u4e0ey\u7684\u5206\u5e03\uff08\u521d\u6b65\u4f30\u8ba1x\u4e0ey\u5e94\u8be5\u662f\u72ec\u7acb\u7684\uff09
\u7136\u540e\u6c42x2\u4e0ey2\u7684\u5206\u5e03 \u7531\u4e8ex\u4e0ey\u72ec\u7acb\uff0cx2\u4e0ey2\u4e5f\u72ec\u7acb\uff0c\u5c31\u53ef\u6c42z~N(\uff09\u7684\u671f\u671b\u548c\u65b9\u5dee\u4e86\uff0c\u7136\u540e\u5199\u4f5c\u6982\u7387\u5bc6\u5ea6\u5373\u53ef\u3002
\u671b\u91c7\u7eb3
∫[-∞→x]∫[-∞→y] f(u，v)dvdu=∫[-∞→x]f(u)du∫[-∞→y] f(v)dv
两边分别对x求偏导得：
∫[-∞→y] f(x，v)dv=f(x)∫[-∞→y] f(v)dv
两边分别对y求偏导得：
f(x，y)=f(x)f(y)
希望可以帮到你，不明白可以追问，如果解决了问题，请点下面的"选为满意回答"按钮，谢谢。
X,Y互相独立
设X的密度函数为f(x),Y的密度函数为f(y)
它们的联合密度函数为f(x,y)=f(x)f(y)
f(y,x)=f(y)f(x)=f(x,y)
绛旓細F锛坸,y锛=F(x)F(y)鈭玔-鈭炩啋x]鈭玔-鈭炩啋y] f(u锛寁)dvdu=鈭玔-鈭炩啋x]f(u)du鈭玔-鈭炩啋y] f(v)dv 涓よ竟鍒嗗埆瀵箈姹傚亸瀵煎緱锛氣埆[-鈭炩啋y] f(x锛寁)dv=f(x)鈭玔-鈭炩啋y] f(v)dv 涓よ竟鍒嗗埆瀵箉姹傚亸瀵煎緱锛歠(x锛寉)=f(x)f(y)甯屾湜鍙互甯埌浣狅紝涓嶆槑鐧藉彲浠ヨ拷闂紝濡傛灉瑙ｅ喅浜嗛棶棰橈紝...
绛旓細绛旀涓猴細F锛坅锛宐锛+1-[F1锛坅锛+F2锛坆锛塢鐢变簬F锛坅锛宐锛=P{X鈮锛孻鈮}锛孎1锛坅锛=P{X鈮锛孻锛+鈭瀩锛孎2锛坆锛=P{X锛+鈭烇紝Y鈮}锛岃岋細P{X锛瀉锛孻锛瀊}=P{X锛+鈭烇紝Y锛+鈭瀩-P{X鈮锛孻锛+鈭瀩-P{X锛+鈭烇紝Y鈮}+P{X鈮锛孻鈮} 鈭碢{X锛瀉锛孻锛瀊}=1-F1锛坅锛-...
绛旓細F锛坸锛寉锛=鈭埆f锛坸锛寉锛塪xdy=鈭埆4xydxdy=鈭玿22ydy=x2y2锛庯紙0鈮鈮1锛0鈮鈮1锛変簩缁闅忔満鍙橀噺( X,Y)鐨勬ц川涓嶄粎涓嶺 銆乊 鏈夊叧,鑰屼笖杩樹緷璧栦簬杩欎袱涓殢鏈哄彉閲忕殑鐩镐簰鍏崇郴銆傚洜姝わ紝閫愪釜鍦版潵鐮旂┒X鎴朰鐨勬ц川鏄笉澶熺殑锛岃繕闇灏嗭紙X锛孻锛変綔涓轰竴涓暣浣撴潵鐮旂┒銆
绛旓細璇︾粏杩囩▼鏄紝鎸夌収姒傜巼瀵嗗害鍑芥暟鐨勬ц川锛屾湁鈭(-鈭,鈭)鈭(-鈭,鈭)p(x,y)dxdy=1銆傗埓鈭(0,1)dx鈭(x,1)Axydy=1銆傝岋紝鈭(0,1)dx鈭(x,1)Axydy=A鈭(0,1)xdx鈭(x,1)ydy銆傚張锛屸埆(x,1)ydy=(1-x²)/2锛鈭(A/2)鈭(0,1)x(1-x²)dx=1銆傗埓(A/2)[x²/...
绛旓細鑱斿悎鍒嗗竷鍑芥暟锛氬皢浜岀淮闅忔満鍙橀噺锛圶锛孻锛鐪嬫垚鏄钩闈笂闅忔満鐐圭殑鍧愭爣锛屽垎甯冨嚱鏁癋锛x锛寉锛夊湪锛坸锛寉锛夊鐨勫嚱鏁板煎氨鏄殢鏈虹偣锛圶锛孻锛夎惤鍦ㄥ鍥句互锛坸锛寉锛変负椤剁偣鑰屼綅浜庤鐐瑰乏涓嬫柟鐨勬棤绌风煩褰㈠尯鍩熷唴鐨勬鐜囥
绛旓細绠鍗曡绠椾竴涓嬪嵆鍙紝绛旀濡傚浘鎵绀
绛旓細鎵浠 P锛圶/Y<0锛=P(X>0,Y<0)+P(X<0,Y>0)=P(X>0)P(Y<0)+P(X<0)P(Y>0)=0.5脳0.5+0.5脳0.5 =0.5 浜岀淮闅忔満鍙橀噺( X,Y)鐨勬ц川涓嶄粎涓嶺 銆乊 鏈夊叧,鑰屼笖杩樹緷璧栦簬杩欎袱涓殢鏈哄彉閲忕殑鐩镐簰鍏崇郴銆傚洜姝わ紝閫愪釜鍦版潵鐮旂┒X鎴朰鐨勬ц川鏄笉澶熺殑锛岃繕闇灏嗭紙X锛孻锛変綔涓轰竴涓暣浣撴潵鐮旂┒...
绛旓細杩欏氨鏄痻銆亂鐨勫彇鍊艰寖鍥淬傝姹侾(X<Y)灏辨槸姹俧(x,y)鍦ㄥ煎煙閲屼笖X<Y閮ㄥ垎鐨勪簩閲嶇Н鍒嗐傜敾鍑哄浘鍍忓氨寰堟槑鏄撅紝鎵瑕佹眰鐨勭Н鍒嗗尯鍩熷氨鏄痽=x²銆亂=1涓巠=x鎵鍥存垚鐨勮緝澶х殑閭ｄ釜鍖哄煙銆傚垯P(X<Y) = 鈭(0,1)鈭(-鈭歽,y)f(x,y)dxdy =7/4 鈭(0,1)y(4)+y(5/2)dy = 17/20 ...
绛旓細渚嬪锛氳浜岀淮闅忔満鍙橀噺(X,Y)鐨勬鐜囧瘑搴︿负 f(x,y)=kx(x-y),0 1)瀵筜浠庯紞X鍒癤绉垎銆瀵筙浠0鍒2绉垎 琚Н鍑芥暟KX(X-Y) 鍋氫簩閲嶇Н鍒嗙瓑浜1 姹傚緱K=8 2)f(x,y)=8x(x-y)X鐨勮竟缂樺瘑搴﹀Y浠-X鍒癤绉垎 Y鐨勮竟缂樺瘑搴﹀嚱鏁板X浠0鍒2绉垎 fx(x)=16x^3 fy(y)=64/3-16Y 3锛塒(0 ...
绛旓細3銆佹潯浠跺垎甯冿紝搴旇鍐欐垚 fX(x|Y=y)鑰岄潪f尉(x|畏=y)锛岃〃绀篩=y鐨勬潯浠跺垎甯冿紝鎸夐鐩剰鎬濓紝姝ゅy鐞嗚В涓烘煇涓甯告暟锛屽垯fX(x|Y=y)=f(x,y)/fY(y)=e^(-y)/ye^(-y)=1/y锛沠Y(y)=ye^(-y)闅忔満鍙橀噺Y鐨勮竟娌垮垎甯冦4銆佹潯浠舵鐜囷紝浼煎簲鍐欐垚P锛圶<2|Y<1)锛屼篃鏄Н鍒嗚绠楋細P锛圶<2|Y<1)...
