设f（x,y）是[a,b]上的正值连续函数，D={（x,y）|a<=x<=b,a<=y<=b}试证二重积分f 设连续性随机变量X的一切可能值在区间[a,b]内，其密度函数...
\u8bbef(x)\u662f\u6b63\u503c\u8fde\u7eed\u51fd\u6570,D={(x,y)|a<=x<=b,a<=y<=b},\u5219I=\u222b\u222bD2f(x)/[f(x)+f(y)] dxdy=?
1\u3001a\u2264X\u2264b\uff0c\u6c42\u671f\u671bE\u6709\u4fdd\u5e8f\u6027\uff0c\u8fd9\u662f\u4e2a\u5b9a\u7406\uff0c\u6240\u4ee5E(a)\u2264E(X)\u2264E(b)\uff0c\u7136\u540e\u5e38\u6570\u7684\u671f\u671b\u5f53\u7136\u7b49\u4e8e\u672c\u8eab\uff0cE(a)=a\uff0cE(b)=b\uff0c\u6240\u4ee5E(a)\u2264X\u2264E(b)\u3002
2\u3001\u8fd9\u4e2a\u9700\u8981\u4e00\u4e2a\u6280\u5de7\uff0c\u505a\u53d8\u6362,Y=(X-a)/(b-a)\uff0cY\u8fd9\u4e2a\u53d8\u91cf\u662f\u5728[0,1]\u4e0a\u5206\u5e03\u7684\u3002
D(X)=D(Y)\u00d7(b-a)²=[E(Y²)-E²(Y)]\u00d7(b-a)²\u3002
Y\u22641\u6240\u4ee5Y²\u2264Y\u6240\u4ee5E(Y²)\u2264E(Y)\u6240\u4ee5D(X)\u2264[E(Y)-E²(Y)]\u00d7(b-a)²\u3002
E(Y)-E²(Y)\u5c31\u662fa-a²\u8fd9\u79cd,a-a²=a(1-a)\u7528\u5747\u503c\u4e0d\u7b49\u5f0fa(1-a)\u2264(a+1-a)²/4=1/4\u3002
\u6240\u4ee5D(X)\u22641/4\u00d7(b-a)²=(b-a)²/4\u5c31\u8bc1\u5b8c\u4e86\u3002
\u6269\u5c55\u8d44\u6599\uff1a
\u6570\u5b66\u671f\u671b\u7684\u6765\u6e90\uff1a
\u572817\u4e16\u7eaa\uff0c\u6709\u4e00\u4e2a\u8d4c\u5f92\u5411\u6cd5\u56fd\u8457\u540d\u6570\u5b66\u5bb6\u5e15\u65af\u5361\u6311\u6218\uff0c\u7ed9\u4ed6\u51fa\u4e86\u4e00\u9053\u9898\u76ee\uff1a\u7532\u4e59\u4e24\u4e2a\u4eba\u8d4c\u535a\uff0c\u4ed6\u4eec\u4e24\u4eba\u83b7\u80dc\u7684\u673a\u7387\u76f8\u7b49\uff0c\u6bd4\u8d5b\u89c4\u5219\u662f\u5148\u80dc\u4e09\u5c40\u8005\u4e3a\u8d62\u5bb6\uff0c\u4e00\u5171\u8fdb\u884c\u4e94\u5c40\uff0c\u8d62\u5bb6\u53ef\u4ee5\u83b7\u5f97100\u6cd5\u90ce\u7684\u5956\u52b1\u3002
\u5f53\u6bd4\u8d5b\u8fdb\u884c\u5230\u7b2c\u56db\u5c40\u7684\u65f6\u5019\uff0c\u7532\u80dc\u4e86\u4e24\u5c40\uff0c\u4e59\u80dc\u4e86\u4e00\u5c40\uff0c\u8fd9\u65f6\u7531\u4e8e\u67d0\u4e9b\u539f\u56e0\u4e2d\u6b62\u4e86\u6bd4\u8d5b\uff0c\u5206\u914d\u8fd9100\u6cd5\u90ce\u6bd4\u8f83\u516c\u5e73\u7684\u65b9\u6cd5\uff1a\u7528\u6982\u7387\u8bba\u7684\u77e5\u8bc6\uff0c\u4e0d\u96be\u5f97\u77e5\uff0c\u7532\u83b7\u80dc\u7684\u53ef\u80fd\u6027\u5927\uff0c\u4e59\u83b7\u80dc\u7684\u53ef\u80fd\u6027\u5c0f\u3002
\u56e0\u4e3a\u7532\u8f93\u6389\u540e\u4e24\u5c40\u7684\u53ef\u80fd\u6027\u53ea\u6709(1/2)\u00d7(1/2\uff09=1/4\uff0c\u4e5f\u5c31\u662f\u8bf4\u7532\u8d62\u5f97\u540e\u4e24\u5c40\u7684\u6982\u7387\u4e3a1\uff0d(1/4)=3/4\uff0c\u7532\u670975%\u7684\u671f\u671b\u83b7\u5f97100\u6cd5\u90ce\uff1b\u800c\u4e59\u671f\u671b\u8d62\u5f97100\u6cd5\u90ce\u5c31\u5f97\u5728\u540e\u4e24\u5c40\u5747\u51fb\u8d25\u7532\uff0c\u4e59\u8fde\u7eed\u8d62\u5f97\u540e\u4e24\u5c40\u7684\u6982\u7387\u4e3a(1/2)*(1/2)=1/4\uff0c\u5373\u4e59\u670925%\u7684\u671f\u671b\u83b7\u5f97100\u6cd5\u90ce\u5956\u91d1\u3002
\u53ef\u89c1\uff0c\u867d\u7136\u4e0d\u80fd\u518d\u8fdb\u884c\u6bd4\u8d5b\uff0c\u4f46\u4f9d\u636e\u4e0a\u8ff0\u53ef\u80fd\u6027\u63a8\u65ad\uff0c\u7532\u4e59\u53cc\u65b9\u6700\u7ec8\u80dc\u5229\u7684\u5ba2\u89c2\u671f\u671b\u5206\u522b\u4e3a75%\u548c25%\uff0c\u56e0\u6b64\u7532\u5e94\u5206\u5f97\u5956\u91d1\u7684100*75%=75(\u6cd5\u90ce)\uff0c\u4e59\u5e94\u5206\u5f97\u5956\u91d1\u7684\u7684100\u00d725%=25(\u6cd5\u90ce)\u3002\u8fd9\u4e2a\u6545\u4e8b\u91cc\u51fa\u73b0\u4e86\u201c\u671f\u671b\u201d\u8fd9\u4e2a\u8bcd\uff0c\u6570\u5b66\u671f\u671b\u7531\u6b64\u800c\u6765\u3002
定积分可随便换积分变量
=∫[a→b] f(x)dx∫[a→b] 1/f(y)dy
=∫∫(D) f(x)/f(y) dxdy 其中：D为a≤x≤b,a≤y≤b
该积分区域为正方形区域,关于y=x对称,则满足轮换对称性,即：∫∫ f(x)/f(y)dxdy=∫∫ f(y)/f(x)dxdy
=(1/2)∫∫(D) [f(x)/f(y) + f(y)/f(x)] dxdy
由平均值不等式
≥∫∫(D) 1 dxdy 被积函数为1,积分结果是区域面积
=(b-a)²
=右边
绛旓細宸﹁竟=鈭[a鈫b] f(x)dx鈭玔a鈫抌] 1/f(x)dx 瀹氱Н鍒嗗彲闅忎究鎹㈢Н鍒嗗彉閲 =鈭玔a鈫抌] f(x)dx鈭玔a鈫抌] 1/f(y)dy =鈭埆(D) f(x)/f(y) dxdy 鍏朵腑锛欴涓篴鈮鈮,a鈮鈮 璇ョН鍒嗗尯鍩熶负姝ｆ柟褰㈠尯鍩,鍏充簬y=x瀵圭О,鍒欐弧瓒宠疆鎹㈠绉版,鍗筹細鈭埆 f(x)/f(y)dxdy=鈭埆 f(y)/f(x)...
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绛旓細鑻ュ嚱鏁皔=f(x)鍦ㄥ尯闂[a,b]涓婄殑鍥捐薄鏄繛缁笉涓鏉℃洸绾裤傛剰鎬濆氨鏄锛氳繖鏄簩娆″嚱鏁帮紝f(a)>0锛宖(b)>0锛岄偅(a,b)涔嬮棿鍙兘娌℃湁闆剁偣銆俧锛a锛f锛坆锛夛紴0锛屾湁鍙兘瀛樺湪瀹炴暟c鈭堬紙a锛宐锛変娇寰梖锛坈锛=0
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绛旓細lim(x鈫y) f(y) = lim(x鈫抷) f(x)鍗筹細f(y) =lim(x鈫抷) f(x)涓寮忓[a,b]閮芥垚绔嬶紝鎵浠ヨ繛缁紒2锛夋牴鎹鎰忥細|f(x)-f(y)|/|x-y|鈮[|x-y|^(k-1)]鈭 -M[|x-y|^(k-1)]鈮f(x)-f(y)]/(x-y)鈮[|x-y|^(k-1)]褰搆>1鏃讹紝瀵逛笂寮忓彇x鈫抷鐨勬瀬闄愶紝鏍规嵁...
