  The time-fractional diffusion-wave equation is revisited, where the time
derivative is of order $2 \nu$ and $0 < \nu \le 1$. The behaviour of the
equation is "diffusion-like" (respectively, "wave-like") when $0 < \nu \le
\frac{1}{2}$ (respectively, $\frac{1}{2} < \nu \le 1$). Two types of
time-fractional derivatives are considered, namely the Caputo and
Riemann-Liouville derivatives. Initial value problems and initial-boundary
value problems are investigated and handled in a unified way using an embedding
method. A two-parameter auxiliary function is introduced and its properties are
investigated. The time-fractional diffusion equation is used to generate a new
family of probability distributions, and that includes the normal distribution
as a particular case.
