  This paper is concerned with the fractionalized diffusion equations governing
the law of the fractional Brownian motion $B_H(t)$. We obtain solutions of
these equations which are probability laws extending that of $B_H(t)$. Our
analysis is based on McBride fractional operators generalizing the hyper-Bessel
operators $L$ and converting their fractional power $L^{\alpha}$ into
Erd\'elyi--Kober fractional integrals. We study also probabilistic properties
of the r.v.'s whose distributions satisfy space-time fractional equations
involving Caputo and Riesz fractional derivatives. Some results emerging from
the analysis of fractional equations with time-varying coefficients have the
form of distributions of time-changed r.v.'s.
