The time-fractional diffusion-wave equation is revisited, where the time derivative is of order��2 �� and 0 < ����� 1. The behaviour of the equation is ���diffusion-like��� (respectively, ���wave-like���) when 0<�����12 (respectively, 12<�����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 studied 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.