概率论:(坐等!) 设X,Y均为标准化随机变量,且有ρ(XY)=1/2,令Z1=aX,Z2=bX+cY.试确定a,b,c的值,使得D(Z1)=D(Z2)=1,且Z1,Z2不相关.a=+-1,b=1/(根号3),c=-2/根号3 或 a=+-1,b=-1/根号3,c=2/根号3
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![概率论:(坐等!) 设X,Y均为标准化随机变量,且有ρ(XY)=1/2,令Z1=aX,Z2=bX+cY.试确定a,b,c的值,使得D(Z1)=D(Z2)=1,且Z1,Z2不相关.a=+-1,b=1/(根号3),c=-2/根号3 或 a=+-1,b=-1/根号3,c=2/根号3](/uploads/image/z/7143993-9-3.jpg?t=%E6%A6%82%E7%8E%87%E8%AE%BA%EF%BC%9A%EF%BC%88%E5%9D%90%E7%AD%89%21%EF%BC%89+%E8%AE%BEX%2CY%E5%9D%87%E4%B8%BA%E6%A0%87%E5%87%86%E5%8C%96%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%2C%E4%B8%94%E6%9C%89%CF%81%EF%BC%88XY%EF%BC%89%3D1%2F2%2C%E4%BB%A4Z1%3DaX%2CZ2%3DbX%2BcY.%E8%AF%95%E7%A1%AE%E5%AE%9Aa%2Cb%2Cc%E7%9A%84%E5%80%BC%2C%E4%BD%BF%E5%BE%97D%EF%BC%88Z1%EF%BC%89%3DD%EF%BC%88Z2%EF%BC%89%3D1%2C%E4%B8%94Z1%2CZ2%E4%B8%8D%E7%9B%B8%E5%85%B3.a%3D%2B-1%2Cb%3D1%2F%EF%BC%88%E6%A0%B9%E5%8F%B73%EF%BC%89%2Cc%3D-2%2F%E6%A0%B9%E5%8F%B73+%E6%88%96+a%3D%2B-1%2Cb%3D-1%2F%E6%A0%B9%E5%8F%B73%2Cc%3D2%2F%E6%A0%B9%E5%8F%B73)
概率论:(坐等!) 设X,Y均为标准化随机变量,且有ρ(XY)=1/2,令Z1=aX,Z2=bX+cY.试确定a,b,c的值,使得D(Z1)=D(Z2)=1,且Z1,Z2不相关.a=+-1,b=1/(根号3),c=-2/根号3 或 a=+-1,b=-1/根号3,c=2/根号3
概率论:(坐等!) 设X,Y均为标准化随机变量,且有ρ(XY)=1/2,令Z1=aX,Z2=bX+cY.
试确定a,b,c的值,使得D(Z1)=D(Z2)=1,且Z1,Z2不相关.
a=+-1,b=1/(根号3),c=-2/根号3 或 a=+-1,b=-1/根号3,c=2/根号3
概率论:(坐等!) 设X,Y均为标准化随机变量,且有ρ(XY)=1/2,令Z1=aX,Z2=bX+cY.试确定a,b,c的值,使得D(Z1)=D(Z2)=1,且Z1,Z2不相关.a=+-1,b=1/(根号3),c=-2/根号3 或 a=+-1,b=-1/根号3,c=2/根号3
∵D(Z1)=D(aX)=E((aX)^2)-(E(aX))^2=a^2*E(X^2)-0=a^2=1
∴a=+-1
∵D(Z2)=D(bX+cY)=E((bX+cY)^2)-(E(bX+cY))^2=b^2*E(X^2)+2bc*E(XY)+c^2*E(Y^2)-0=b^2+2bc*E(XY)+c^2
又∵ρ(XY)=1/2,∴E(XY)=1/2,∴D(Z2)=b^2+bc+c^2=1 ①
∵Z1,Z2不相关,∴cov(Z1,Z2)=E(Z1Z2)-E(Z1)*E(Z2)=E(ab*X^2+ac*XY)-0=0
即ab+1/2*ac=0,b+1/2*c=0 ②
联立①②,解得b=1/(根号3),c=-2/根号3或b=-1/根号3,c=2/根号3
综上a=+-1,b=1/(根号3),c=-2/根号3 或 a=+-1,b=-1/根号3,c=2/根号3
very easy a ~ ∵X,Y均为标准化随机变量,∴ D(x)=D(Y)=1;
D(z1)=(a^2)D(x)=1,∴a^2=1;a=+1或-1;
D(z2)=(b^2)D(x)+(c^2)D(Y)+2bc cov(x,y);
=(b^2)+(c^2)+2bc cov(x,y) ( *式子)其中 cov(x,y)=E(x,y)-E(x)E...
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very easy a ~ ∵X,Y均为标准化随机变量,∴ D(x)=D(Y)=1;
D(z1)=(a^2)D(x)=1,∴a^2=1;a=+1或-1;
D(z2)=(b^2)D(x)+(c^2)D(Y)+2bc cov(x,y);
=(b^2)+(c^2)+2bc cov(x,y) ( *式子)其中 cov(x,y)=E(x,y)-E(x)E(y);
∵p(x,y)=1/2=(cov(x,y))/(√D(x)√D(Y));
∴cov(x,y)=1/2;带入(*式子);
D(z2)=b^2+c^2+bc=1;这个可以用配方法解答
b^2+cb+c^2-1=0
b^2+cb+(c^2)/4+3(c^2)/4-1=0
(b+c/2)^2+3(c^2)/4-1=0
∴b=-c/2;c=+2/(√3)或-2/√3;可得答案
就是上面的答案,望采纳~~~
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