求二维随机变量的密度函数。如：（1） （X，Y）是区域D上的均匀分布，D为单位圆的第一象限部分，求
\u4e8c\u7ef4\u968f\u673a\u53d8\u91cf\uff08X,Y)\u5728\u533a\u57dfD={(x,y)|0<x<1,0<y<x^2}\u4e0a\u670d\u4ece\u5747\u5300\u5206\u5e03\uff0c\u6c42\uff08X,Y)\u8054\u5408\u5bc6\u5ea6\u51fd\u6570\u8fd8\u662f\u4f60\u554a\u3002
\u7b54\u6848\u5982\u56fe\uff0c\u5e0c\u671b\u4f60\u80fd\u770b\u61c2\u3002
好难
绛旓細鍥犳锛屽彉閲廦 鐨勬鐜囧瘑搴﹀嚱鏁癴(Y) 涓篎(Y) 鐨勫鏁般傚F(Y)姹傚寰楋細f(Y) = dF(Y)/dY = 1/2, (1<Y<3)鍥犳锛屽湪 (1,3) 鍖洪棿鍐咃紝闅忔満鍙橀噺 Y=2X+1 鐨勬鐜瀵嗗害鍑芥暟 f1(Y) = 1/2銆傚湪鍏朵粬鍖洪棿鍐咃紝姒傜巼瀵嗗害鍑芥暟涓洪浂銆
绛旓細璁浜岀淮闅忔満鍙橀噺(X,Y)鐨勫瘑搴﹀嚱鏁涓篺锛坸,y锛=Ae^-(2x+3y),x>0,y>0,f锛坸,y锛=0,鍏朵粬 姒傜巼P锛圶澶т簬Y锛変负A/6銆傛鐜嘝锛濃埆鈭玣锛坸锛寉锛塪xdy 锛滱鈭玡锛撅紙锛2x锛塪x鈭玡锛撅紙锛3y锛塪y 锛滱锛婏蓟锛2e锛撅紙锛2x锛夛冀锝滐紙0锛岋紜鈭烇級锛婏蓟锛3e锛撅紙锛3y锛夛冀锝滐紙0锛岋紜鈭烇級锛滱锛6 ...
绛旓細锛1锛夋眰Z = max{X, Y}鐨勫瘑搴﹀嚱鏁帮細1. 鎴戜滑闇瑕佹壘鍑篫 = max{X, Y}鍦ㄤ笉鍚屽尯鍩熶笂鐨勫垎甯冩儏鍐点2. 褰揦 鈮 Y鏃讹紝Z = Y锛涘綋X > Y鏃讹紝Z = X銆3. 鏍规嵁瀵嗗害鍑芥暟 f(x, y) 鐨勫畾涔夊煙锛屾垜浠彲浠ュ垎涓轰袱涓儏鍐垫潵璁＄畻锛歛. 褰1 - x 鈮 y 鈮 1 涓 0 鈮 x 鈮 1 鏃讹紝Z = Y锛屽洜姝...
绛旓細=1-(arcsiny)/蟺 Y=sinX鐨勬鐜瀵嗗害鍑芥暟f(y)=F(y)'=-1/[蟺鈭(1-y^2)] ,0<y<=1;f(y)=0,y鍦ㄥ叾瀹冭寖鍥淬俋~U(0,蟺)锛堝潎鍖鍒嗗竷锛夛紝0<cosX<=1,瀵逛簬-1<y<1,Y=cosX鐨勫垎甯冨嚱鏁癋(y)=P(Y<=y)=P(cosx<=y)=P(arccosy<=x<蟺)=(蟺-arccosy)/蟺=1-(arccosy)/蟺 Y=cosX...
绛旓細+ x²)∂F(x, y)/∂y = (1 + arctan(2y)) / 2 鍐嶅涓婅堪涓や釜鍋忓鏁拌繘琛屾眰瀵硷紝鎴戜滑寰楀埌锛∂²F(x, y)/∂x∂y = 1 / (2(1 + x²))鍥犳锛浜岀淮闅忔満鍙橀噺(X, Y)鐨勬鐜瀵嗗害鍑芥暟涓猴細f(x, y) = 1 / (2(1 + x²))
绛旓細^2=4-8=-4 E(Y)=2鈭(-鈭,2]ydy-鈭(-鈭,2]y²dy=2[y²/2]^2-(y³/3)^2=4-8=-4 鍥犳锛孹鐨勬湡鏈涘间负-4锛孻鐨勬湡鏈涘间篃涓-4銆傜患涓婃墍杩帮紝浜岀淮杩炵画鍨闅忔満鍙橀噺(X,Y)鐨勮仈鍚瀵嗗害鍑芥暟涓簆(x,y)=2-x-y锛宲(x,y)=0鏃讹紝X鐨勬湡鏈涘间负-4锛孻鐨勬湡鏈涘间篃涓-4銆
绛旓細灏浜岀淮鍒嗗竷鍑芥暟姹娣峰悎浜岄樁鍋忓,灏卞緱鍒瀵嗗害鍑芥暟.濡涓嬪浘锛
绛旓細缁间笂鎵杩帮紝Z鐨勫瘑搴﹀嚱鏁颁负锛歠(z) = { (1-z)e^z, 0 鈮 z 鈮 1 0, z > 1 } (2) 姹俍=min{X,Y}鐨勫瘑搴﹀嚱鏁帮細瀵逛簬Z=min{X,Y}锛屾垜浠彲浠ラ氳繃璁＄畻鍏剁疮绉垎甯冨嚱鏁版潵姹傝В鍏跺瘑搴﹀嚱鏁般傞鍏堬紝鎴戜滑鍙互璁＄畻Z鐨凜DF锛屽嵆P(Z鈮)銆傚綋z<0鏃讹紝P(Z鈮)=0锛屽洜涓篫鐨勫彇鍊艰寖鍥存槸闈炶礋鏁般傚綋0...
绛旓細濡傚浘锛屽湪骞抽潰鐩磋鍧愭爣绯讳腑锛岀洿绾縴=4/3+4鍒嗗埆浜杞达紝y杞翠簬A锛孊涓ょ偣锛岀偣C涓篛B鐨勪腑鐐癸紝鐐笵鍦ㄧ浜岃薄闄愶紝涓斿洓杈瑰舰AOCD涓虹煩褰.锛1锛夌洿鎺ュ啓鍑虹偣A锛孊鍧愭爣锛屽苟姹傜洿绾緼B涓嶤D浜ょ偣鐨勫潗鏍囷紱锛2锛夊姩鐐筆浠庣偣C鍑哄彂锛屾部绾挎CD浠ユ瘡绉1涓崟浣嶉暱搴︾殑閫熷害鍚戠粓鐐笵杩愬姩锛涘悓鏃跺姩鐐筂浠庣偣A鍑哄彂锛屾部鐩寸嚎AB浠ユ瘡绉5/3...
绛旓細鍋囪鏈変袱涓繛缁瀷闅忔満鍙橀噺X鍜孻锛屽叾鑱斿悎姒傜巼瀵嗗害鍑芥暟涓篺(x,y)銆傚畾涔夊畠浠殑鍜屽嚱鏁颁负Z=X+Y锛岄珮鍑芥暟涓篧=X/Y銆傜幇鍦ㄩ渶瑕佹眰Z鍜學鐨勬鐜囧瘑搴﹀嚱鏁般傞鍏堟眰Z鐨勬鐜囧瘑搴﹀嚱鏁般傚彲浠ヤ娇鐢ㄥ嵎绉叕寮忔潵姹傝В銆傛牴鎹嵎绉叕寮忥紝Z鐨勬鐜囧瘑搴﹀嚱鏁癴_Z(z)鍙互琛ㄧず涓猴細f_Z(z) = 鈭玣(x, z-x)dx 鍏朵腑锛寈鐨勫彇鍊艰寖鍥...
