正態比率分佈

X/Y

X

Y

sigma_x

sigma_y

rho=0

U=X/Y

P(u)

=

int_(-infty)^infty|y|f(uy,y)dy

=

1/(2pisigma_xsigma_y)int_(-infty)^infty|y|e^(-[y^2/(2sigma_y^2)+u^2y^2/(2sigma_x^2)])dy

=

1/(pisigma_xsigma_y)int_0^inftyyexp[-y^2(1/(2sigma_y^2)+(u^2)/(2sigma_x^2))]dy.

P(u)

=

1/(pisigma_xsigma_y)1/(2(1/(2sigma_y^2)+(u^2)/(2sigma_x^2)))

=

1/pi(sigma_xsigma_y)/(u^2sigma_y^2+sigma_x^2)

=

1/pi((sigma_x)/(sigma_y))/(u^2+((sigma_x)/(sigma_y))^2),

P_(X/Y)(u)

=

int_(-infty)^inftyint_(-infty)^infty(e^(-x^2/(2sigma_x^2)))/(sigma_xsqrt(2pi))(e^(-y^2/(2sigma_y^2)))/(sigma_ysqrt(2pi))delta(x/y-u)dxdy

=

1/pi((sigma_x)/(sigma_y))/(u^2+((sigma_x)/(sigma_y))^2),

delta(x)