  We study the distribution of residence time or equivalently that of ``mean
magnetization" for a family of Gaussian Markov processes indexed by a positive
parameter $\alpha$. The persistence exponent for these processes is simply
given by $\theta=\alpha$ but the residence time distribution is nontrivial. The
shape of this distribution undergoes a qualitative change as $\theta$
increases, indicating a sharp change in the ergodic properties of the process.
We develop two alternate methods to calculate exactly but recursively the
moments of the distribution for arbitrary $\alpha$. For some special values of
$\alpha$, we obtain closed form expressions of the distribution function.
