联合密度函数

根据联合密度函数，求协方差

E(XY)=∫(-∞,+∞)∫(-∞,+∞)xy(x+y)dxdy =∫(0,1)∫(0,1)xy(x+y)dxdy=∫(0,1)[(1/3)y+(1/2)y^2]dy=1/3。 E(X)=∫(-∞,+∞)∫(-∞,+∞)x(x+y)dxdy=∫(0,1)∫(0,1)x(x+y)dxdy=7/12 E(Y)==∫(0,1)∫(0,1)y(x+y)dxdy=7/12 cov(X,Y)=E(XY)-E(X)E(Y)=1/3-49/144=-1/144。 E(X^2)=∫(0,1)∫(0,1)x^2(x+y)dxdy=5/12, E(Y^2)=∫(0,1)∫(0,1)y^2(x+y)dxdy...全部

  E(XY)=∫(-∞,+∞)∫(-∞,+∞)xy(x+y)dxdy =∫(0,1)∫(0,1)xy(x+y)dxdy=∫(0,1)[(1/3)y+(1/2)y^2]dy=1/3。 E(X)=∫(-∞,+∞)∫(-∞,+∞)x(x+y)dxdy=∫(0,1)∫(0,1)x(x+y)dxdy=7/12 E(Y)==∫(0,1)∫(0,1)y(x+y)dxdy=7/12 cov(X,Y)=E(XY)-E(X)E(Y)=1/3-49/144=-1/144。
   E(X^2)=∫(0,1)∫(0,1)x^2(x+y)dxdy=5/12, E(Y^2)=∫(0,1)∫(0,1)y^2(x+y)dxdy=5/12, D(X)=E(X^2)-E(X)^2=5/12-49/144=11/144。
   D(Y)=11/144。 pXY=cov(X,Y)/[√D(X)√D(Y)]=-1/11。收起