设二维随机变量(X,Y)在区域G={(x,y)|0

题目

设二维随机变量(X,Y)在区域G={(x,y)|0

答案

cxysxsgwhm77766041542011-09-24 22:59:06vxjfjghuncx0df(x,y)=2 E(X)=∫[-1,0]dx∫[-1-x,0]2xdy =∫[-1,0]2x(1+x)dx=(x^2+2/3*x^3)|[-1,0]=-1/3 同理：E(Y)=-1/3 E(XY)=∫[-1,0]dx∫[-1-x,0]2xydy =∫[-1,0]xy^2|[-1-x,0]dx=-∫[-1,0]x(1+x)^2dx =-(1/4*x^4+2/3*x^3+1/2*x^2)|[-1,0]=1/12 COV(X,Y)=E(XY)-EX*EY=-1/36 E(X^2)=∫[-1,0]dx∫[-1-x,0]2x^2dy =∫[-1,0]2x^2(1+x)dx=(2/3*x^3+1/2*x^4)|[-1,0]=1/6 D(X)=E(X^2)-(EX)^2=1/18 同理：D(Y)=1/18