随机变量的均匀分布证明!!急!!! 设随机变量X服从区间(-2,2)上的均匀分布,设随机变量Y=...
\u8bbe\u968f\u673a\u53d8\u91cfu\u670d\u4ece(\u20142,2)\u7684\u5747\u5300\u5206\u5e03,\u968f\u673a\u53d8\u91cfx=\u20141,\u5728\u7ebf\u7b49\u6025\u6025\u6025\u6025\u6025\u6025\u4f60\uff01\uff01\uff01\uff01\uff01\uff01\uff01\uff01\uff01\uff01\uff011\u7b54\uff1a
\u8bbeX,Y\u76f8\u4e92\u72ec\u7acb\uff0c\u4e14\u670d\u4ece\u540c\u5206\u5e03X\uff5eU(-2,2),Y\uff5eU(-2,2),
\u5219X,Y\u7684\u6982\u7387\u5bc6\u5ea6\u4e3a(y\u53ea\u9700\u6362\u6210x)
f(x):
\u2460:1/4,-2<x<2,
\u2461:0\uff0c\u5176\u5b83\uff0c
\u7531\u5377\u79ef\u516c\u5f0f\uff0c
fZ(z)=\u222bfX(x)fY(z-x)dx (\u5176\u4e2d\u79ef\u5206\u4e0a\u9650\u4e3az+2\uff0c\u4e0b\u9650\u4e3az-2,\u5728\u5750\u6807\u7cfb\u91cc\u753b\u51fa-2\u2264x\u22642,-2\u2264z-x\u22642\u7684\u56fe\u50cf\uff09
=\u222b(1/4*1/4)dx
=1/4
\u6545\u5f97Z=X+Y\u5728\u56fe\u793a\u7684\u533a\u57dfG\u91cc\u5747\u5300\u5206\u5e03,
\u7528(x.y)\u8868\u793a\u533a\u57df\u91ccG\u7684\u70b9,\u5219
f(x,y):
\u2460:1/4,(x,y)\u2208G
\u2461:0,\u5176\u5b83\uff0c
\u6240\u4ee5Z\u7684\u5206\u5e03\u51fd\u6570\u4e3aF(z):
\u2460:0,z\u2264-4,
\u2461:(z+4)^2/8,-4<z<0,
\u2462:1-1/8(4-z)^2,0\u2264z<4,
\u2463:1.z\u22654,
Z\u7684\u6982\u7387\u5bc6\u5ea6\u4e3af(z):
\u2460\uff1az/4+1,-4<z<0,
\u2461:1-z/4,0\u2264z<4,
\u2462:0,\u5176\u5b83\u3002
\u671b\u91c7\u7eb3
\u4f60\u597d\uff01\u56e0\u4e3aEX=\u222b(-2\u52302)x(1/4)dx=0\uff0cEY=\u222b(-2\u52302)(x^6)(1/4)dx=64/7\uff0cE(XY)=\u222b(-2\u52302)(x^7)(1/4)dx=0\u3002\u6240\u4ee5cov(X,Y)=E(XY)-(EX)(EY)=0\uff0c\u5373X\u4e0eY\u4e0d\u76f8\u5173\u3002\u7ecf\u6d4e\u6570\u5b66\u56e2\u961f\u5e2e\u4f60\u89e3\u7b54\uff0c\u8bf7\u53ca\u65f6\u91c7\u7eb3\u3002\u8c22\u8c22\uff01
如果按你的意思就是 F(b)-F(a)=g(b-a),其中函数g(x)的具体表达式不知道, 且定义域为(-1,1)因为F(b)-F(a)=g(b-a)恒成立,故可将a、b看做是两个变量
对b求偏导有:f(b)=g'(b-a) (1)
注意:(1)式对于任意b、a均成立,所以可以固定b,变动a
那我们发现左边始终不变,所以右边的值随a也不变,也就是g'(b-a)=常数,并在整个定义域都成立
即f(b)=C,也即概率密度为常数,也就是均匀分布
绛旓細濡傛灉U鏄紙0,1锛変笂鐨勫潎鍖鍒嗗竷鐨勫彉閲 鍒橮( U < y ) = y 鎵浠-1(u)鐨勫垎甯冩槸 P锛團-1(U) < y )= P 锛圲 < F(y)锛 = F(y) 涓哄垎甯僃 鍏朵腑F-1(x)鏄弧瓒矲(y)=x鐨刌鍊,鎶 y=F-1(x) 甯﹀叆 F(y)寰 F(y)= F锛團-1(x)锛=x ...
绛旓細鍗砯(b)=C,涔熷嵆姒傜巼瀵嗗害涓哄父鏁帮紝涔熷氨鏄鍧囧寑鍒嗗竷
绛旓細棣栧厛闇瑕佺‘瀹鍧囧寑鍒嗗竷鐨鍒嗗竷鍑芥暟鐨勫畾涔夊煙銆傚浜庤繛缁闅忔満鍙橀噺X锛屽叾瀹氫箟鍩熶负[a锛 b]銆傝绠楁鐜囧瘑搴﹀嚱鏁 鍧囧寑鍒嗗竷鐨勬鐜囧瘑搴﹀嚱鏁颁负f锛坸锛 = 1/锛坆-a锛夛紝鍏朵腑a涓哄畾涔夊煙鐨勪笅闄愶紝b涓哄畾涔夊煙鐨勪笂闄愩傛帹瀵煎垎甯冨嚱鏁 鏍规嵁姒傜巼瀵嗗害鍑芥暟鐨勫畾涔夛紝鍙互鎺ㄥ鍑哄潎鍖鍒嗗竷鐨勫垎甯冨嚱鏁癋锛坸锛 = Prob锛圶 鈮 x锛夈傚綋x ...
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绛旓細绠鍗曡绠椾竴涓嬪嵆鍙紝绛旀濡傚浘鎵绀
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绛旓細since Xhas a uniform distribution pdf of fX锛坸锛塱s fX(x)=1 x鈭圼0,1] fX锛坸锛=0 otherwise FY(y)=p(Y鈮)=P锛坅X+b鈮锛塱f a锛0, then FY锛坹锛=p锛圶鈮わ紙y-b锛/a锛=FX锛堬紙y-b锛/a锛塪ifferentiating FY(y)yields fY...
绛旓細X鏈嶄粠鍧囧寑鍒嗗竷锛屽嵆X~U(a锛宐),鍒橢(X)=(a+b)/2, D(X)=(b-a)²/12 璇佹槑濡備笅锛氳杩炵画鍨闅忔満鍙橀噺X~U(a锛宐)閭d箞鍏跺垎甯冨嚱鏁癋(x)=(x-a)/(b-a),a鈮鈮 E(x)=鈭獸(x)dx=鈭(a鍒癰)(x-a)/(b-a)dx =(x²/2-a)/(b-a) |(a鍒癰)=(b²/2-a)/(b-...
绛旓細銆愮瓟妗堛戯細鍥燜(x)鍗曡皟銆佽繛缁紝涓斿彇鍊间簬[0锛1]锛屾晠褰搚锛0鏃讹紝P(畏鈮)=0锛庡綋y锛1鏃讹紝P(畏鈮)=1锛庡綋0鈮鈮1鏃讹紝鏈塒(畏鈮)=P(F(尉)鈮)=P(尉鈮-1(y))=F[F-1(y)]=y鏁呂锋槸(0锛1)涓鐨勫潎鍖鍒嗗竷锛
绛旓細鐢卞凡鐭闅忔満鍙橀噺X~U(2,4)锛屽彲浠ユ眰鍑篨鐨勬鐜囧瘑搴﹀嚱鏁颁负锛歠(x) = 1/(4-2) = 1/2, 2 鈮 x 鈮 4 鍥犳锛孹鏄竴涓鍧囧寑鍒嗗竷鐨勯殢鏈哄彉閲忥紝鍙互鏍规嵁鍧囧寑鍒嗗竷鐨勬湡鏈涘拰鏂瑰樊鍏紡姹傚嚭E(X)鍜孌(2X+2)銆傛眰E(X)锛欵(X) = (2 + 4) / 2 = 3 姹侱(2X+2)锛氶鍏堟眰D(2X)锛屾牴鎹柟宸殑鎬ц川锛...