= Solution
{c}
Differentiate the <cumulative distribution function> for $y>0$, using $\Phi'(z)=(2\pi)^{-1/2}e^{-z^2/2}$ and $d\sqrt{2y}/dy=1/\sqrt{2y}$. The <probability density function> is
$$
\boxed{f_Y(y)=\begin{cases}\dfrac{e^{-y}}{\sqrt{\pi y}},&y>0,\\0,&y<0.\end{cases}}
$$
Its value at zero can be assigned arbitrarily without changing any <probability>. Although this <probability density function> diverges as $y\downarrow0$, $F_Y(0)=0$ and there is no <point mass> at zero. Equivalently, $Y$ has a <gamma distribution> with shape $1/2$ and rate one.
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