概率密度f(x,y)=4xye*-(x*2+y*2),x大于0,y大于0,其他为0,求根号下x*

f(x,y)=4xye^(-(x²+y²)),x>0,y>0; 其它为0.
Z=√(X²+Y²),x=rcosθ,y=rsinθ,dxdy=rdzdθ
F(z)=P(Z≤z)=∫∫f(x,y)dxdy=∫[-π,π]{∫[0,z]f(rcosθ,rsinθ)rdr}dθ
f(z)=dF(z)/dz=∫[-π,π]{f(zcosθ,zsinθ)z}dθ
此题中,x,y均为正,故θ的值域为[0,π/2].
f(z)=dF(z)/dz=∫[-π,π]{f(zcosθ,zsinθ)z}dθ
f(z)=∫[0,π/2]{f(zcosθ,zsinθ)z}dθ
=4(z^3)e^(-z^2)∫[0,π/4]cosθsinθdθ
=4(z^3)e^(-z^2)(1/2) = 2(z^3)e^(-z^2)
E{√(X²+Y²)}=E{Z}=∫[0,∞]2(z^4)e^(-z^2)dz
=(3/4)(√π)≈1.32934