已知二维随机变量（x,y）的概率密度函数为f（x,y）＝ae∧-（x+y）,x＞0,y＞0 0，其他
\u8bbe\u4e8c\u7ef4\u968f\u673a\u53d8\u91cf(X,Y)\u7684\u6982\u7387\u5bc6\u5ea6\u4e3af(x,y)={e^-y,0<x<y;0,\u5176\u4ed6.}\u6c42Z=X+Y1\u3001\u6c42\u968f\u673a\u53d8\u91cfX\u7684\u5bc6\u5ea6fX(x)\uff0c\u8fb9\u6cbf\u5206\u5e03
fX(x)={e^(-y)\uff1b0<x<y\uff1b{0
2\u3001\u6982\u7387\u5bc6\u5ea6\u51fd\u6570f(x,y)\u5728\u76f4\u7ebfx=0,y=x,y=-x+1\u6240\u56f4\u7684\u4e09\u89d2\u5f62\u533a\u57df\u7684\u4e8c\u91cd\u5ea6\u79ef\u5206\uff0c\u7ed3\u679c\u662f1+e^(-1)-2e^(-1/2)
3\u3001\u6761\u4ef6\u5206\u5e03\uff0c\u5e94\u8be5\u5199\u6210 fX(x|Y=y)\u800c\u975ef\u03be(x|\u03b7=y)\uff0c\u8868\u793aY=y\u7684\u6761\u4ef6\u5206\u5e03\uff0c\u6309\u9898\u76ee\u610f\u601d\uff0c\u6b64\u5904y\u7406\u89e3\u4e3a\u67d0\u4e00\u5e38\u6570\uff0c\u5219fX(x|Y=y)=f(x,y)/fY(y)=e^(-y)/ye^(-y)=1/y\uff1bfY(y)=ye^(-y)\u968f\u673a\u53d8\u91cfY\u7684\u8fb9\u6cbf\u5206\u5e03\u3002
4\u3001\u6761\u4ef6\u6982\u7387\uff0c\u4f3c\u5e94\u5199\u6210P\uff08X<2|Y<1)\uff0c\u4e5f\u662f\u79ef\u5206\u8ba1\u7b97\uff1aP\uff08X<2|Y<1)\uff0c=P{X<2,Y<1}/P(Y<1)
P{X<2,Y<1}\u4e3af(x,y)\u5728\u76f4\u6743\u7ebfx=2,y=1,y=x\u6240\u56f4\u533a\u57df\u79ef\u5206\uff0cP(Y<1)\u4e3af(x,y)\u5728\u76f4\u7ebfy=x,y=1\u6240\u56f4\u533a\u57df\u79ef\u5206\uff0c\u5728\u672c\u9898\u60c5\u51b5\uff0c\u4e24\u4e2a\u533a\u57df\u7684\u6709\u6548\u90e8\u5206\uff08\u5373\u4e0d\u4e3a\u96f6\u90e8\u5206\uff09\u6070\u597d\u76f8\u7b49\uff0c\u6545\u79ef\u5206\u503c\u4e3a1\u3002\u6982\u7387\u610f\u4e49\u662f\uff0c\u968f\u673a\u70b9\u5206\u5e03\u533a\u57df\u4e3a0<x<y\uff0c\u6709Y<1\uff0c\u5219\u5fc5\u6709X<2\u77e3\u3002
\u4f8b\u5982\uff1a
\u2235P(X>2\u4e28Y2,Y2,Y<4)\u3001P(Y<4)\u5373\u53ef\u5f97\u3002
\u800c\uff0cP(X>2,Y<4)=\u222b(2,4)dy\u222b(2,y)f(x,y)dx=\u222b(2,4)(y-2)e^(-y)dy=-(y-1)e^(-y)\u4e28(y=2,4)=e^(-2)-3e^(-4)\u3002
\u5bf9P(Y0\u3001fY(y)=0,y\u4e3a\u5176\u5b83\u3002\u2234P(Y<4)=\u222b(0,4)fY(y)dy=\u222b(0,4)ye^(-y)dy=-(y+1)e^(-y)\u4e28(y=0,4)=1-5e^(-4)\u3002
\u2234P(X>2\u4e28Y2,Y<4)/P(Y<4)=[e^(-2)-3e^(-4)]/[1-5e^(-4)]\u3002
\u6269\u5c55\u8d44\u6599\uff1a
\u4e8c\u7ef4\u968f\u673a\u53d8\u91cf( X,Y)\u7684\u6027\u8d28\u4e0d\u4ec5\u4e0eX \u3001Y \u6709\u5173\uff0c\u800c\u4e14\u8fd8\u4f9d\u8d56\u4e8e\u8fd9\u4e24\u4e2a\u968f\u673a\u53d8\u91cf\u7684\u76f8\u4e92\u5173\u7cfb\u3002\u56e0\u6b64\uff0c\u9010\u4e2a\u5730\u6765\u7814\u7a76X\u6216Y\u7684\u6027\u8d28\u662f\u4e0d\u591f\u7684\uff0c\u8fd8\u9700\u5c06\uff08X\uff0cY\uff09\u4f5c\u4e3a\u4e00\u4e2a\u6574\u4f53\u6765\u7814\u7a76\u3002
\u4e00\u822c\uff0c\u8bbeE\u662f\u4e00\u4e2a\u968f\u673a\u8bd5\u9a8c\uff0c\u5b83\u7684\u6837\u672c\u7a7a\u95f4\u662fS={e}\uff0c\u8bbeX=X\uff08e\uff09\u548cY=Y(e)S\u662f\u5b9a\u4e49\u5728S\u4e0a\u7684\u968f\u673a\u53d8\u91cf\uff0c\u7531\u5b83\u4eec\u6784\u6210\u7684\u4e00\u4e2a\u5411\u91cf\uff08X\uff0cY\uff09\uff0c\u53eb\u505a\u4e8c\u7ef4\u968f\u673a\u53d8\u91cf\u6216\u4e8c\u7ef4\u968f\u673a\u5411\u91cf\u3002
\u6709\u4e00\u4e2a\u73ed\uff08\u5373\u6837\u672c\u7a7a\u95f4\uff09\u4f53\u68c0\u6307\u6807\u662f\u8eab\u9ad8\u548c\u4f53\u91cd\uff0c\u4ece\u4e2d\u4efb\u53d6\u4e00\u4eba\uff08\u5373\u6837\u672c\u70b9\uff09\uff0c\u4e00\u65e6\u53d6\u5b9a,\u90fd\u6709\u552f\u4e00\u7684\u8eab\u9ad8\u548c\u4f53\u91cd\uff08\u5373\u4e8c\u7ef4\u5e73\u9762\u4e0a\u7684\u4e00\u4e2a\u70b9\uff09\u4e0e\u4e4b\u5bf9\u5e94\uff0c\u8fd9\u5c31\u6784\u9020\u4e86\u4e00\u4e2a\u4e8c\u7ef4\u968f\u673a\u53d8\u91cf\u3002\u7531\u4e8e\u62bd\u6837\u662f\u968f\u673a\u7684,\u76f8\u5e94\u7684\u8eab\u9ad8\u548c\u4f53\u91cd\u4e5f\u662f\u968f\u673a\u7684,\u6240\u4ee5\u8981\u7814\u7a76\u5176\u5bf9\u5e94\u7684\u5206\u5e03\u3002
\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1-\u4e8c\u7ef4\u968f\u673a\u53d8\u91cf
fx(x)=\u222b(0~) xe^(-x-y) dy
=xe^(-x) (x>0)
=0 \u5176\u4ed6x
fy(y)=\u222b(0~) xe^(-x-y) dx
=e^(-y) (y>0)
(\u222b(0~)xe^(-x) dx =1 \u8fd9\u4e2a\u6839\u636e\u4f3d\u9a6c\u51fd\u6570\u5f88\u5bb9\u6613\u7b97,\u222b(0~) t^(n) e^(-t) dt=n!)
=0 \u5176\u4ed6y
\u6269\u5c55\u8d44\u6599\uff1a
\u6982\u7387\u6307\u4e8b\u4ef6\u968f\u673a\u53d1\u751f\u7684\u673a\u7387\uff0c\u5bf9\u4e8e\u5747\u5300\u5206\u5e03\u51fd\u6570\uff0c\u6982\u7387\u5bc6\u5ea6\u7b49\u4e8e\u4e00\u6bb5\u533a\u95f4(\u4e8b\u4ef6\u7684\u53d6\u503c\u8303\u56f4)\u7684\u6982\u7387\u9664\u4ee5\u8be5\u6bb5\u533a\u95f4\u7684\u957f\u5ea6\uff0c\u5b83\u7684\u503c\u662f\u975e\u8d1f\u7684\uff0c\u53ef\u4ee5\u5f88\u5927\u4e5f\u53ef\u4ee5\u5f88\u5c0f\u3002
\u5bf9\u4e8e\u968f\u673a\u53d8\u91cfX\u7684\u5206\u5e03\u51fd\u6570F\uff08x\uff09\uff0c\u5982\u679c\u5b58\u5728\u975e\u8d1f\u53ef\u79ef\u51fd\u6570f(x)\uff0c\u4f7f\u5f97\u5bf9\u4efb\u610f\u5b9e\u6570x\uff0c\u5219X\u4e3a\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\uff0c\u79f0f(x)\u4e3aX\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\uff0c\u7b80\u79f0\u4e3a\u6982\u7387\u5bc6\u5ea6\u3002
\u5b9a\u7406\uff1a\u8bbe\u968f\u673a\u53d8\u91cfX\u5177\u6709\u6982\u7387\u5bc6\u5ea6fX(x)\uff0c-\u221e0\uff08\u6216\u6052\u6709g'(x)<0\uff09\uff0c\u5219Y=g(X)\u662f\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\u3002
2. e^(-x) ,-$<x<+$
3. 1-e^[(1/2)*x].
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绛旓細鍙埄鐢ㄨ仈鍚堟鐜囧瘑搴︾殑浜岄噸绉垎涓1锛屾眰鍑簁=2銆傝竟闄呭瘑搴﹀嚱鏁扮殑姹傝В锛屾湰璐ㄥ氨鏄冨療绉垎锛屽彧瑕佽浣忚竟缂樻鐜囧瘑搴﹀氨鏄鑱斿悎瀵嗗害鍑芥暟姹傜Н鍒嗭紝褰撴垜浠眰鍏充簬Y鐨勮竟闄呭瘑搴﹀嚱鏁版椂灏辨槸瀵逛簬f锛坸锛寉锛夌殑鑱斿悎瀵嗗害鍑芥暟鍏充簬X姹傜Н鍒嗭紝姹俌鐨勮竟闄呭瘑搴﹀嚱鏁板垯鍚岀悊銆傜浜岄儴鍒嗘槸姹闅忔満鍙橀噺鍑芥暟鐨勫瘑搴︼紝涓鑸敤鍒嗗竷鍑芥暟娉曪紝鍗冲厛鐢...
绛旓細鑻涓嶻鐩镐簰鐙珛锛屽垯f(x,y)=fx(x) * fy(y)鍗宠仈鍚堟鐜囧瘑搴︾瓑浜巟鍜寉杈圭紭瀵嗗害鐨勪箻绉 鏄剧劧鍦ㄨ繖閲 0鈮鈮鈮1锛宖x(x)=鈭(0鍒1) f(x,y) dy =鈭(0鍒1) 8xy dy =4x²y (浠ｅ叆y鐨勪笂涓嬮檺1鍜0)=4x²鍚岀悊鍙互寰楀埌fy(y)=4y²锛屾墍浠 fx(x) * fy(y)=4x² ...
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绛旓細浜岀淮闅忔満鍙橀噺(X锛孻)鏄竴涓潎鍖鍒嗗竷锛屽潎鍖鍒嗗竷姹傛鐜囷紝鍙渶闈㈢Н鐩搁櫎灏辫浜 P(X+Y<=1)=(1*1/2)/2=1/4
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