已知二维随机变量(X,Y)的概率密度为 f(x,y)={ x^2+1/3xy 0<=x<=1,0<=y<=2 , 0 其他} 则X的概率密度fx(x)=
\u4e8c\u7ef4\u968f\u673a\u53d8\u91cf(X,Y)\u7684\u6982\u7387\u5bc6\u5ea6\u4e3a f(x,y)={ x^2+1/3xy\u6c42Y<0.5 \u7684\u6982\u7387,\u5148\u786e\u5b9ax\u548cy\u6240\u56f4\u6210\u7684\u533a\u57df\uff0c\u7136\u540e\u753b\u51fay=0.5\u4e0e\u533a\u57df\u6240\u56f4\u6210\u7684\u65b0\u7684\u533a\u57df\uff0c\u7136\u540e\u76f4\u63a5\u7b97\u8be5\u533a\u57df\u9762\u79ef\u3002\u5373\u7b97\u4e8c\u91cd\u79ef\u5206\uff0c\u5176\u4e2dx\u7684\u4e0a\u4e0b\u9650\u662f0\u52301\uff0cy\u7684\u662fx\u52300.5
fx(x)=∫(0->2) x^2+1/3xy dy=2x^2+2/3x 0<=x<=1 其他0fx(x)=∫(0->2) x^2+1/3xy dy=2x^2+2/3x 0<=x<=1 其他0
绛旓細鏍规嵁姒傜巼鐨勫姞娉曞叕寮忥細 P(X+Y) = P(X) + P(Y) - P(XY) 鍙緱 鈶犲綋0<x<1,0<y<2x 鏃讹紝fz(z) = fx(X) + fy(Y) - f(x,y) = 2x+1-1 = 2x 鈶犲綋x涓巠涓嶈兘鍚屾椂婊¤冻 0<x<1,0<y<2x 鏃讹紝fz(z) = fx(X) + fy(Y) - f(x,y) = ...
绛旓細fx(x)=鈭(0->2) x^2+1/3xy dy=2x^2+2/3x 0<=x<=1 鍏朵粬0
绛旓細fx(x)=鈭(0->2) x^2+1/3xy dy=2x^2+2/3x 0
绛旓細fy(y)鏄x姹傜Н鍒嗭紝鐩稿綋浜庝簩閲嶇Н鍒嗕腑鐨刌鍨嬶紝鍨傜洿y杞寸殑鍖洪棿鍐呯敾涓鏉＄嚎锛屽彲鐭ワ紝x鐨勭Н鍒嗕笂涓嬮檺鏀瑰彉浜嗭紝涓嶆槸瀵瑰簲涓鏍风殑涓婁笅闄愶紝鎵浠ヨ鍒嗘銆傚綋o<y<1鏃,fy(y)=鍦▂鍒嗕箣涓鍒版鏃犵┓瀵箈姹傜Н鍒 褰1鈮<+鈭炴椂,fy(y)锛濆湪y鍒版鏃犵┓瀵箈姹傜Н鍒嗐傚綋y<0鏃讹紝鐢变簬f(x锛寉)=0锛屾湁fy(y)=0....
绛旓細P=鈭(0-->1)e^(-y)dy鈭(0-->1-y)2e^(-2x)dx =鈭(0-->1)e^(-y)(1-e^(2(1-y))dy =鈭(0-->1)(e^(-y)-e^2e^y)dy=(1-e)(1+e^2)
绛旓細褰0鈮鈮1锛0鈮鈮1鏃 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锛浜岀淮闅忔満鍙橀噺( X,Y)鐨鎬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1锛塮(x,y)=1,0<x<1,0<y<1,=0,鍏朵粬 锛2锛夌浉浜掔嫭绔嬨俧X(x)=3x^2 fY(y)=4y^3 f(x,y)=fX(x)*fY(y) 鎵浠ワ細鐙珛
绛旓細鍖栫疮娆＄Н鍒嗭紝鍏堝y绉垎銆傚乏杈箉绉垎绾 0鍒x锛x绉垎绾0鍒1/2銆傚彸杈箉绉垎绾0鍒1-x,x绉垎绾1/2鍒1銆
绛旓細1. a = 1 2. e^(-x) ,-$<x<+ 3. 1-e^[(1/2)*x].
