设二维随机变量(X, Y)在区域D=｛(x，y)|0＜x＜1，|y|＜x｝内服从均匀分布，求(X, 设二维连续型随机变量(X,Y)在区域D=={(x,y)|0<...

\u8bbe\u4e8c\u7ef4\u968f\u673a\u53d8\u91cf(X\uff0cY)\u5728\u533a\u57dfD={(x\uff0cy)|0\uff1cx\uff1c1\uff0e|y|\uff1cx}\u5185\u670d\u4ece\u5747\u5300\u5206\u5e03\uff0e \u6c42\uff1a(1

\u89e3\uff1a\u2235D={(x,y)\u4e280<x<1\uff0c-x<y<x}\uff0c\u2234D\u662fy=x\u3001y=-x\u3001x=1\u56f4\u6210\u7684\u4e09\u89d2\u533a\u57df\uff0c\u5176\u9762\u79efSD=1*2/2=1\u3002
\u2234\u6309\u7167\u5747\u5300\u5206\u5e03\u7684zhi\u5b9a\u4e49\uff0c\uff08x,y\uff09\u7684\u5bc6\u5ea6\u51fd\u6570\u4e3adaof(x,y)=1/SD=1\uff0c(x,y)\u2208D\u3001f(x,y)=0\uff0c(x,y)∉D\u3002
(1)fX(x)=\u222b(-\u221e,\u221e)f(x,y)dy=\u222b(-x,x)dy=2x\uff0c\u5176\u4e2d0<x<1\u3002fY(y)=\u222b(-\u221e,\u221e)f(x,y)dx=\u222b(0,1)dx=1\uff0c\u5176\u4e2d-1<y<1\u3002
(2)\u2235fX(x)*fY(y)=2x\u2260f(x,y)\uff0c\u2234X\u3001Y\u4e0d\u76f8\u4e92\u72ec\u7acb\u3002
(3)\u22350<x<1\uff0c-x<y<x\uff0c\u22340<x+y<2x\uff0c\u53730<z<2x\uff0c0<x<1\u3002\u2234F(Z=z)=P(0<z<2x)\u3002
\u2234\u7531\u5bc6\u5ea6\u8ba1\u7b97\u516c\u5f0f\uff0c\u6709fZ(z)=F'(Z=z)=fX(z)*[dx/dz]=z/2\uff0c\u5176\u4e2d0<z<2\u3002
\u968f\u673a\u53d8\u91cf\u7684\u6027\u8d28
\u968f\u673a\u53d8\u91cf\u5728\u4e0d\u540c\u7684\u6761\u4ef6\u4e0b\u7531\u4e8e\u5076\u7136\u56e0\u7d20\u5f71\u54cd\uff0c\u53ef\u80fd\u53d6\u5404\u79cd\u4e0d\u540c\u7684\u503c\uff0c\u6545\u5176\u5177\u6709\u4e0d\u786e\u5b9a\u6027\u548c\u968f\u673a\u6027\uff0c\u4f46\u8fd9\u4e9b\u53d6\u503c\u843d\u5728\u67d0\u4e2a\u8303\u56f4\u7684\u6982\u7387\u662f\u4e00\u5b9a\u7684\uff0c\u6b64\u79cd\u53d8\u91cf\u79f0\u4e3a\u968f\u673a\u53d8\u91cf\u3002\u968f\u673a\u53d8\u91cf\u53ef\u4ee5\u662f\u79bb\u6563\u578b\u7684\uff0c\u4e5f\u53ef\u4ee5\u662f\u8fde\u7eed\u578b\u7684\u3002
\u5982\u5206\u6790\u6d4b\u8bd5\u4e2d\u7684\u6d4b\u5b9a\u503c\u5c31\u662f\u4e00\u4e2a\u4ee5\u6982\u7387\u53d6\u503c\u7684\u968f\u673a\u53d8\u91cf\uff0c\u88ab\u6d4b\u5b9a\u91cf\u7684\u53d6\u503c\u53ef\u80fd\u5728\u67d0\u4e00\u8303\u56f4\u5185\u968f\u673a\u53d8\u5316\uff0c\u5177\u4f53\u53d6\u4ec0\u4e48\u503c\u5728\u6d4b\u5b9a\u4e4b\u524d\u662f\u65e0\u6cd5\u786e\u5b9a\u7684\uff0c\u4f46\u6d4b\u5b9a\u7684\u7ed3\u679c\u662f\u786e\u5b9a\u7684\uff0c\u591a\u6b21\u91cd\u590d\u6d4b\u5b9a\u6240\u5f97\u5230\u7684\u6d4b\u5b9a\u503c\u5177\u6709\u7edf\u8ba1\u89c4\u5f8b\u6027\u3002\u968f\u673a\u53d8\u91cf\u4e0e\u6a21\u7cca\u53d8\u91cf\u7684\u4e0d\u786e\u5b9a\u6027\u7684\u672c\u8d28\u5dee\u522b\u5728\u4e8e\uff0c\u540e\u8005\u7684\u6d4b\u5b9a\u7ed3\u679c\u4ecd\u5177\u6709\u4e0d\u786e\u5b9a\u6027\uff0c\u5373\u6a21\u7cca\u6027\u3002

\u89e3\u7b54\u8fc7\u7a0b\u4e0e\u7ed3\u679c\u5982\u56fe\u6240\u793a\u3002

简单计算一下即可，答案如图所示

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