Past exam of the Probability, statistics and options II course of the University of Oxford / 2010 / M / M4 / a / iii / Solution

Assume $a,b>0$, the standard deviations. Substitute the two normal densities into (ii):The last step uses ${\int }_0^{\infty }u{e}^{-A{u}^2/2}du=1/A$. With $c=b/a$ this is $\boxed{f_V(v)=c/[\pi (c^2+v^2)]}$, the centered Cauchy density of scale $c$.

The desired event is $|V|<1$, outside the null event $X=0$. Integrating the density gives