设二维连续型随机向量（x,y）的概率密度为f(x,y)={2ye-x,x>0,0<y<1,0其他，求：边缘密度函数？ 设二位随机向量(X,Y)的概率密度函数为f(x,y)=2-x...
\u8bbe\u4e8c\u7ef4\u968f\u673a\u53d8\u91cf\uff08X\uff0cY\uff09\u7684\u6982\u7387\u5bc6\u5ea6\u4e3af\uff08x\uff0cy\uff09=1\uff0c0\uff1cx\uff1c1\uff0c0\uff1cy\uff1c2x0\uff0c \u5176\u4ed6\uff0e\u6c42\uff1a\uff08\u2160\uff09\uff08X\uff0cY\uff09\u7684\u8fb9\u7f18\uff08 I\uff09\u6c42\u5173\u4e8eX\u7684\u8fb9\u9645\u5bc6\u5ea6\u51fd\u6570\u65f6\u5c31\u662f\u5bf9\u4e8ef\uff08x\uff0cy\uff09\u7684\u8054\u5408\u5bc6\u5ea6\u51fd\u6570\u5173\u4e8eY\u6c42\u79ef\u5206\uff0c\u6240\u4ee5\uff1a\u5173\u4e8eX\u7684\u8fb9\u7f18\u6982\u7387\u5bc6\u5ea6fx\uff08x\uff09=\u222b+\u221e?\u221ef(x\uff0cy)dy=\u222b2x0dy\uff0c0\uff1cx\uff1c10 \uff0c \u5176\u4ed6=2x\uff0c0\uff1cx\uff1c10 \uff0c \u5176\u4ed6 \u5173\u4e8eY\u7684\u8fb9\u7f18\u6982\u7387\u5bc6\u5ea6fy\uff08y\uff09=\u222b+\u221e?\u221ef(x\uff0cy)dx=\u222b1y2dx\uff0c0\uff1cy\uff1c20\uff0c \u5176\u4ed6=1?y2\uff0c 0\uff1cy\uff1c20\uff0c \u5176\u4ed6\uff08\u2161\uff09\u4ee4FZ\uff08z\uff09=P{Z\u2264z}=P{2X-Y\u2264z}\uff081\uff09\u5f53z\uff1c0\u65f6\uff0cFZ\uff08z\uff09=P{2X-Y\u2264z}=0\uff1b\uff082\uff09\u5f530\u2264z\uff1c2\u65f6\uff0cFZ\uff08z\uff09=P{2X-Y\u2264z}=?0\uff1cx\uff1cz2\uff0c0\uff1cy\uff1c2xdxdy+?z2\uff1cx\uff1c1\uff0c2x?z\uff1cy\uff1c2xdxdy=z?14z2\uff1b\uff083\uff09\u5f53z\u22652\u65f6\uff0cFZ\uff08z\uff09=P{2X-Y\u2264z}=1\uff1b\u5373\u5206\u5e03\u51fd\u6570\u4e3a\uff1aFZ(z)\uff1d0\uff0c z\uff1c0z?14z2\uff0c0\u2264z\uff1c21 z\u22652\uff0c\u6545\u6240\u6c42\u7684\u6982\u7387\u5bc6\u5ea6\u4e3a\uff1afz(z)\uff1d1?12z\uff0c0\uff1cz\uff1c20 \uff0c \u5176\u4ed6\uff08\u2162\uff09P{Y\u226412|X\u226412}=P(X\u226412\uff0cY\u226412)P(X\u226412)=?y2\uff1cx\uff1c12\uff0c0\uff1cy\uff1c12dxdy?0\uff1cx\uff1c12\uff0c0\uff1cy\uff1c2xdxdy=31614=34\uff0e
\u89e3\uff1a\u4f7f\u7528\u5377\u79ef\u516c\u5f0f
f(z)=\u222b(-\u221e,+\u221e)f(x,z-x)dx
= z(2-z), 0 =< z < 1;
= (2-z)^2, 1 =< z< 2;
= 0, \u5176\u4ed6
0<z<1\u65f6\uff0c\u56e0\u4e3a 0<y<1, 0<z-x<1, \u6240\u4ee5 0<x<z.
f(z) = \u222b (-\u221e,+\u221e) f(x,z-x)dx
= \u222b(0,z) (2-z)dx
= z(2-z), 0<z<1.
\u5f531 x > z-1,
f(z) = \u222b (-\u221e,+\u221e) f(x,z-x)dx
= \u222b (z-1,1) (2-z)dx = (2-z)^2, 1 =< z < 2.
根据变量的取值范围
对联合概率密度函数积分
对y积分得到X的边缘概率密度
对x积分得到Y的边缘概率密度
过程如下：
绛旓細f Y( y)=2e^-(2y),y>0鏃讹紝0锛涘叾瀹冩椂 f (x, y)=f X(x)*f Y( y)锛岀嫭绔 P{ 0<X鈮1锛0<Y鈮2}=锛1-1/e^3锛夛紙1-1/e^4锛夊亣璁捐繖浜涘熀鏈殑闅忔満浜嬩欢鍙戠敓鐨勬鐜囬兘鏄浉绛夌殑锛屽鏋滄湁n涓熀鏈殑闅忔満浜嬩欢锛岃浣垮緱鍙戠敓鐨勬鐜囦箣鍜屼负1銆
绛旓細瀵y绉垎寰楀埌X鐨勮竟缂樻鐜囧瘑搴 瀵x绉垎寰楀埌Y鐨勮竟缂樻鐜囧瘑搴 杩囩▼濡備笅锛
绛旓細姹X锛孻钀藉湪鏌愬尯鍩熺殑姒傜巼灏辨槸璁＄畻姒傜巼瀵嗗害鍦ㄨ繖涓尯鍩熶笂鐨勪簩閲嶇Н鍒嗐傜瓟妗堢瓑浜0.5,0.25銆傝В锛歅(X<= 0.5)灏辨槸鍥句腑褰撳皬-1<=x<2杩欎竴娈电殑鏃跺欙紙鍥犱负-1<=0.5<2锛夛紝鎵浠ュ氨灏0.5浠ｅ叆杩欎竴娈电殑鍒嗗竷鍑芥暟灏卞彲浠ヤ负0.25锛汸(1.5<=X <=2.5) = P(X<=2.5)-P(X<1.5)P(X<=2.5) = ...
绛旓細E(x)=xfX(x)浠0-1绉垎 寰楀嚭2/5 E(xy)=xyf(x,y)鍏堢НY锛屼粠0-2(1-X)鍚庣НX锛屼粠0-1锛屾渶鍚庡緱鍑4/15銆
绛旓細褰搝锛0鏃讹紝FZ锛坺锛=P锛圸鈮锛=P锛圶+Y鈮锛= 鈭埆 x+y鈮f(x锛寉)dxdy = 鈭 z 2 0dx 鈭 z−x xxe−ydy锛 鈭 ,2,璁句簩缁撮殢鏈哄悜閲忥紙X锛孻锛鑱斿悎姒傜巼瀵嗗害涓篺锛坸锛寉锛= x e −y 锛0锛渪锛測 0锛屽叾瀹 锛屾眰锛氾紙1锛夋潯浠舵鐜囧瘑搴 Y|X锛y|x锛夛紱锛...
绛旓細浜岀淮杩炵画闅忔満鍚戦噺x锛寉涓嶄竴瀹氭槸杩炵画鐨勩俋鏄杩炵画鍨嬮殢鏈鍙橀噺锛孻鐨勫畾涔夋槸褰揦=0鏃讹紝Y=1.鍒橸鏄疿鐨勫嚱鏁帮紝Y鍙湁涓や釜鍙栧硷紝鏄鏁ｅ瀷鐨勩
绛旓細灏辨槸涓涓Н鍒嗭細1銆佸厛纭畾A =1/9,2,鍐嶆眰P{(X,Y)鈭圖}=1/9鈭埆锛(6-x-y)dxdy=8/27
绛旓細x^2鈮 杩欎釜鏉′欢鏄粷瀵硅婊¤冻鐨 y鐨勫彇鍊煎彈鍒朵簬x鐨勫彇鍊 杩欓噷x鑼冨洿 鏄 0 1 鎵浠ョН鍒唝鐨勮寖鍥存槸 x^2 鍒 x x绉垎鑼冨洿鏄 0 1 瀵规鐜囧嚱鏁扮Н鍒 寰桟=6
绛旓細璁 鏄簩缁撮殢鏈哄悜閲忥紝瀵逛簬浠绘剰瀹炴暟x锛寉锛岀О浜屽厓鍑芥暟 涓 鐨勫垎甯冨嚱鏁 鎬ц川 浜岀淮绂绘暎鍨嬮殢鏈哄悜閲 瀹氫箟3.2.1 璁句簩缁绂绘暎鍨嬮殢鏈哄悜閲 鎵鏈夊彲鑳界殑鍙栧间负 鏄剧劧鏈夛細浜岀淮杩炵画鍨嬮殢鏈哄悜閲 瀹氫箟3.3.1 瀵逛簬浜岀淮闅忔満鍚戦噺 涓哄叾鍒嗗竷鍑芥暟锛岃嫢瀛樺湪闈炶礋鍑芥暟 浣垮緱瀵逛换鎰忓疄鏁皒,y鎬绘湁 鍒欑О(X,Y)鏄簩缁磋繛缁...
绛旓細瑙ｏ細鈶犲X锛孻鐨勫彉鍖栬寖鍥翠负锛-1鈮鈮1锛-鈭(1-x²)鈮鈮も垰(1-x²)銆傗埓X鐨勮竟缂樺瘑搴X(x)=鈭(-鈭,鈭)f(x,y)dy=(2/蟺)鈭(1-x²)锛屽叾涓瓁鈭(-1,1)銆傚悓鐞嗭紝Y鐨勭殑杈圭紭瀵嗗害fY(y)=鈭(-鈭,鈭)f(x,y)dx=(2/蟺)鈭(1-y²)锛屽叾涓瓂鈭(-1,1)銆傗埓E(...
