设二维随机变量(X，Y)在三角形区域D：0≤y≤x≤1上服从均匀分布，求它的联合分布函数．
【答案】：当x≥1，0≤y＜1时，F(x，y)=y(2-y)；当0≤x＜1，y≥x时，F(x，y)=x2；当0≤y＜x＜1时，F(x，y)=y(2x-y)．绛旓細X~U(0,2蟺)鍒嗗竷鍑芥暟F(y)=P(y)=P(Y<=y)=P(cosx)<=y)=P(arccosy<=x<=2蟺-arccosy)=(1/2蟺)(2蟺-arccosy--arccosy)=(1/2蟺)(2蟺-2arccosy)=1-arccosy/蟺 姒傜巼瀵嗗害鍑芥暟f(y)=F(y)'=1/(蟺(1-y^2)^(1/2)) -1<=y<=1 ...
绛旓細闅忔満鍙橀噺(X,Y)鍦鍖哄煙D 鏈嶄粠鍧囧寑鍒嗗竷,鍒欒仈鍚堝瘑搴﹀嚱鏁癙(X,Y)=1/惟,惟=1/2鍗冲尯鍩烡鐨勯潰绉,涓虹洿绾縳=0,y=x,y=1鎵鍥寸殑閮ㄥ垎,鎵浠(X,Y)=2,10,鍥犲叾涓哄潎鍖鍒嗗竷锛屽垯P(X,Y)涓轰竴甯告暟銆傚張P(X,Y)婊¤冻锛氣埆鈭玃(X,Y)dXdY=1锛岃P(X,Y)=k 锛屽垯鏈夆埆鈭玃(X,Y)dXdY=k鈭埆1dXdY=1 锛屸埆...
绛旓細鈶″浜庡瘑搴﹀嚱鏁帮紝鍏堝x绉垎锛屽啀瀵箉绉垎锛堥兘鏄彉涓婇檺绉垎锛屼粠-鈭炵Н鍒嗗埌x鍜寉锛夛紝鍙互寰楀埌鍒嗗竷鍑芥暟銆傚洜姝ょ煡閬撳垎甯冨嚱鏁帮紝姹傚瘑搴﹀嚱鏁帮紝鍙鍏堝x姹傚锛屽啀瀵箉姹傚灏卞彲浠ャ備篃灏辨槸 f(x,y)=partial² F(x,y)/(partial x partial y)锛堝厛瀵箈姹傚亸瀵兼暟鍐嶅y姹傚亸瀵兼暟锛屾槸浜岄樁娣峰悎鍋忓鏁帮級妤间富鍐...
绛旓細鍙堣闈簡鍝...鐜板湪浣犲簲璇ヤ細鍋氫簡鍚...= = 瑙佷笅鍥惧惂
绛旓細浜岀淮闅忔満鍙橀噺锛圶锛孻锛钀藉湪闀挎柟褰㈠尯鍩燂細宸︿笅瑙掞紙x1锛寉1锛夆啋鍙充笂瑙掞紙x2锛寉2锛夌殑姒傜巼涓猴細F(x2锛寉2)-F(x2锛寉1)-F(x1锛寉2)+F(x1锛寉1)锛屼功涓婃湁杩欏叕寮忋傛棰樹腑锛屽尯鍩燂經0锛渪鈮ο/4锛屜/6锛測鈮ο/3锝濆寘鍚湪锝0锛渪鈮ο/2锛0锛測鈮ο/2锝濅腑锛屾墍浠ュ彲浠ヤ娇鐢 F(x锛寉)=sinxsiny璁＄畻...
绛旓細绗竴涓瓑鍙锋槸鑱斿悎鍒嗗竷鍑芥暟涓庤仈鍚堝瘑搴﹀嚱鏁颁箣闂寸殑鍏崇郴锛屼粠杩炵画鍨闅忔満鍙橀噺鑱斿悎鍒嗗竷鍑芥暟鐨勫畾涔変腑灏卞彲寰楀嚭 绗簩涓瓑鍙峰氨鏄亸瀵兼暟鐨勮绠楋細∂F/∂x=a(c+arctan2y)/(1+x²)∂²F/∂x∂y=a/[(1+x²)(1+4y²)]甯屾湜鍙互甯埌浣狅紝涓嶆槑鐧藉彲浠ヨ拷闂...
绛旓細1銆佹眰闅忔満鍙橀噺X鐨勫瘑搴X(x)锛岃竟娌垮垎甯 fX(x)={e^(-y)锛0<x<y锛泏0 2銆佹鐜囧瘑搴﹀嚱鏁癴(x,y)鍦鐩寸嚎x=0,y=x,y=-x+1鎵鍥寸殑涓夎褰㈠尯鍩熺殑浜岄噸搴︾Н鍒嗭紝缁撴灉鏄1+e^(-1)-2e^(-1/2)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...
绛旓細1銆佹眰闅忔満鍙橀噺X鐨勫瘑搴X(x)锛岃竟娌垮垎甯 fX(x)={e^(-y)锛0<x<y锛泏0 2銆佹鐜囧瘑搴﹀嚱鏁癴(x,y)鍦鐩寸嚎x=0,y=x,y=-x+1鎵鍥寸殑涓夎褰㈠尯鍩熺殑浜岄噸搴︾Н鍒嗭紝缁撴灉鏄1+e^(-1)-2e^(-1/2)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...
绛旓細1銆佹眰闅忔満鍙橀噺X鐨勫瘑搴X(x)锛岃竟娌垮垎甯 fX(x)={e^(-y)锛0<x<y锛泏0 2銆佹鐜囧瘑搴﹀嚱鏁癴(x,y)鍦鐩寸嚎x=0,y=x,y=-x+1鎵鍥寸殑涓夎褰㈠尯鍩熺殑浜岄噸搴︾Н鍒嗭紝缁撴灉鏄1+e^(-1)-2e^(-1/2)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...
绛旓細1銆佹眰闅忔満鍙橀噺X鐨勫瘑搴X(x)锛岃竟娌垮垎甯 fX(x)={e^(-y)锛0<x<y锛泏0 2銆佹鐜囧瘑搴﹀嚱鏁癴(x,y)鍦鐩寸嚎x=0,y=x,y=-x+1鎵鍥寸殑涓夎褰㈠尯鍩熺殑浜岄噸搴︾Н鍒嗭紝缁撴灉鏄1+e^(-1)-2e^(-1/2)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...
