Derive pdf of slash distribution, standard normal divided by standard uniform

Question

Let $U$ be a standard uniform distribution, and $Z$ be a standard normal distribution. $Z$ and $U$ are independent. Let $X=Z/U$, and find the pdf of $X$.

I searched around the internet and all answers used some advanced results in ratio distribution, but I don't think this is what I'm expected to do. I want to know if I can only use ways like CDF or transformation methods to find the pdf of $X$.

Answer 1

So maybe we can use the cdf. Note that $f_U(u) = \mathbb{I}_{[0,1]}$, so: $$ \begin{split} F_X(x) &= \mathbb{P}[X \le x] \\ &= \int_0^1 \mathbb{P}[Z/U \le x|U = u] \cdot f_U(u)\ du \\ &= \int_0^1 \mathbb{P}[Z \le xu] \cdot 1\ du \\ &= \int_0^1 \Phi(xu) du. \end{split} $$ Now differentiate to get the pdf.