Past exam of the year 1 of the computer science course of the University of Oxford / 2015 / Discrete mathematics, continuous mathematics, and probability / 9 / c / i / Solution

The random variable $Y=Z^2/2$ is nonnegative, so its cumulative distribution function is zero for $y<0$. If $y\ge 0$, thenThe last step uses the symmetry of the standard normal distribution, $\Phi (-z)=1-\Phi (z)$. Hence