二维随机向量求边缘概率密度
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绛旓細濡備笅鍥炬墍绀
绛旓細姹傞殢鏈鍙橀噺X鐨勫瘑搴X(x)锛岃竟娌垮垎甯僨X(x)={e^(-y)锛0<x<y锛泏0 姒傜巼瀵嗗害鍑芥暟f(x,y)鍦ㄧ洿绾縳=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)鏉′欢鍒嗗竷锛屽簲璇ュ啓鎴 fX(x|Y=y)鑰岄潪f尉(x|畏=y)锛岃〃绀篩=y鐨勬潯浠跺垎甯锛屾寜棰樼洰鎰忔濓紝姝ゅy鐞嗚В涓烘煇涓甯告暟...
绛旓細,2,璁浜岀淮闅忔満鍚戦噺锛圶锛孻锛夎仈鍚堟鐜囧瘑搴︿负f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锛鏉′欢姒傜巼瀵嗗害f Y|X锛坹|x锛夛紱锛2锛塟=X+Y姒傜巼瀵嗗害锛
绛旓細(1)锛岀敱姒傜巼瀵嗗害鐨勬ц川鍙緱锛屸埆(0,1)dy鈭(-鈭歽,鈭歽)f(x,y)dx=1銆傗埓c鈭(0,1)ydy鈭(-鈭歽,鈭歽)x²dx=1銆傗埓4c/21=1锛宑=21/4銆(2)锛孹銆乊鐨勮竟缂樺垎甯冨瘑搴鍑芥暟鍒嗗埆涓猴紝fX(x)=鈭(x²,1)f(x,y)dy=c鈭(x²,1)x²ydy=鈥=(21/8)x²(1-x^4)...
绛旓細杈圭紭瀵嗗害鍑芥暟鐨勬剰鎬濇槸鎸囪竟缂樺垎甯冨嚱鏁般傝仈鍚堝瘑搴﹀嚱鏁扮敤鍏紡f锛坸锛寉锛=fx锛坸锛塮y锛坹锛夋眰寰銆傝仈鍚堝瘑搴﹀嚱鏁颁害绉板缁村垎甯冨嚱鏁帮紝闅忔満鍚戦噺鐨勫垎甯冨嚱鏁帮紝浠ヤ簩缁存儏褰负渚嬶紝鑻ワ紙X锛孻锛夋槸浜岀淮闅忔満鍚戦噺锛寈銆亂鏄换鎰忎袱涓疄鏁帮紝鍒欑О浜屽厓鍑芥暟銆傝仈鍚堝瘑搴﹀嚱鏁扮殑鎰忎箟锛氬鏋滃皢浜岀淮闅忔満鍙橀噺(X锛孻)鐪嬫垚鏄钩闈笂闅忔満鐐圭殑...
绛旓細濡傛灉浜岀淮闅忔満鍙橀噺X,Y鐨勫垎甯冨嚱鏁癋{x锛寉}涓哄凡鐭ワ紝閭ｄ箞闅忔満鍙橀噺x锛寉鐨勫垎甯冨嚱鏁癋x{x}鍜孎y{y}鍒嗗埆鍙敱F{x锛寉}姹傚緱銆傚垯Fx{x}鍜孎y{y}涓哄垎甯冨嚱鏁癋{x锛寉}鐨勮竟缂樺垎甯冨嚱鏁般杈圭紭瀵嗗害鍑芥暟姹傝В鏂规硶鏄細鏍规嵁鍙橀噺鐨勫彇鍊艰寖鍥达紝瀵硅仈鍚堟鐜囧瘑搴﹀嚱鏁扮Н鍒嗭紝瀵箉绉垎寰楀埌X鐨勮竟缂樻鐜囧瘑搴銆傝竟缂樻鐜囧瘑搴︿篃绉版鐜囧瘑搴...
绛旓細1銆姹傞殢鏈鍙橀噺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銆姒傜巼瀵嗗害鍑芥暟f(x,y)鍦ㄧ洿绾縳=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鐨勬潯浠跺垎甯锛屾寜棰樼洰鎰忔濓紝姝ゅy...
绛旓細f(X,Y)=1/2 fY(y)=1/2y fY(1)=1/2
绛旓細姹俋锛孻钀藉湪鏌愬尯鍩鐨姒傜巼灏辨槸璁＄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) = ...
绛旓細璁闅忔満鍚戦噺(X,Y)鐨勫瘑搴﹀嚱鏁颁负F(x,y)=kxy,0鈮鈮1,0鈮鈮1 0,鍏朵粬 姹傚弬鏁発鐨勫煎強(X,Y)鐨勮竟缂樻鐜囧瘑搴濡備笅锛
