【题目】设随机变量X,Y相互独立,且都服从(0,1)上的均匀分布(1)求E(XY),E(X/Y),E ln(XY)] , E[|Y-X|] .(2)以X,Y为边长作一长方形,以A,C分别表示长方形的面积和周长,求A和C的相关系数

【解析】解(1)X,Y的概率密度都是f (x)=1,0x1;0,①.1,0x1,E(XY)=E(X)E(Y)Y=1/2*1/2=1/4 E[]不存在(因发散)E[ (XY)]=∫_a^1∫_0^1(ln,x+lny)dxdyy=2∫_0^1∫_0^1(lnx)dxdy y=xD=-2.DE(|Y-x|) ox=∫_0^π(|y-x|dxdy) (如题4.25图 D=D_1∪D_2)2( =2∫((y-x)dxdy=2∫_0^1)∫_x^1(y -x)dxdy=2∫_0^1∫_x^1(y-x)dydx=1/2(2) A=XY,C=2(X+β) Y),Cov(A,C)=E(AC)-E(A)E(C).AC=2X^2Y+2XY^2 ,E(X^2)=E(Y^2)=D(X)+(E(X))^2=1/(12)+1/4=1/3 E(AC)=2E(X^2Y)+2E(XY^2) =2E(X^2)E(Y)+2E(X)E(Y^2) )=2*1/3*1/2+2*1/2*1/3=2/3 Cov(A,C)=E(AC)-E(A)E(C)=2/3-[E(X)E(Y)*2(E(X)+E(Y)] =2/3-[1/2*1/2*2(1/2+1/2)]=1/6 D(A)=E(X^2Y^2)-[E(X)E(Y)]^2=E(X^2)E(Y^2)-(1/2)^(-1/2))^(1/(2 =(1/3)^2-(1/4)^2=7/(144) D(C)=D(2X+2Y)=D(2X)+D(2Y)=4*1/(12)+4*1/(12)=2/3 故P_(AC)=(Cov(A_1C))/(√(D(A)D(C)))=1/67√(7/(144)*2/3)=√(6/7)