Let $X$, $Y$ and $Z$ be independent random variables with characteristic function $$\phi(\theta) = \frac{1}{\sqrt{1+\theta^2}}$$
say whether the r.v. $X+Y+Z$ has a continuous density and evaluate $\mathrm{P}(X+Y+Z \ge 0)$.
Since $X$,$Y$ and $Z$ are independent we know that the characteristic function of $U =X+Y+Z$ is $$\phi_U(\theta)=\frac{1}{(1+\theta^2)^{\frac{3}{2}}}$$
which is integrable on $\mathbb{R}$, so $U$ has a continuous density:
$$f(u)=\frac{1}{2 \pi}\int_{- \infty}^{+ \infty}\frac{e^{-i \theta u}}{(1+\theta^2)^{\frac{3}{2}}}d\theta$$
now $$\mathrm{P}(X+Y+Z \ge 0) = \int_{0}^{+ \infty} f(u)du$$
but since we can't have a solution in terms of standard functions of $f$ there is surely another way to do it, any suggestion?