>>804
P(Z<z)=∫[Z<z]f(Z)dZ
P(X^2+Y^2<z)=∬[X^2+Y^2<z]g(X,Y)dXdY
=∬[R^2<z]g(RcosΘ,RsinΘ)RdRdΘ
=∬[R^2<z]Re^-R^2/2/2πdRdΘ
=∫[R^2<z]e^-R^2/2d(R^2/2)
=[-e^-R^2/2][R^2<z]
=1-e^-z/2(z>0),0(z<0)
f(z)=e^-z/2/2(z>0),0(z<0)
P(W<w)=∫[W<w]h(W)dW
P(X^2/(X^2+Y^2)<w)=∬[X^2/(X^2+Y^2)<w]g(X,Y)dXdY
=∬[cos^2Θ<w]g(RcosΘ,RsinΘ)RdRdΘ
=∬[cos^2Θ<w]Re^-R^2/2/2πdRdΘ
=∬[cos^2Θ<w]e^-R^2/2/2πd(R^2/2)dΘ
=∫[cos^2Θ<w][-e^-R^2/2][R>0]dΘ
=∫[cos^2Θ<w]dΘ
＝[Θ][cos^2Θ<w]
=2π-4acos√w(0<w<1),0(w<0),2π(w>1)
h(w)=2/√w(1-w),(0<w<1),0(w<0,>1)