Let $I\subset\mathbb{R}$ be an interval and $T_a:[0,1]\to[0,1]$, $a\in I$, a
one-parameter family of piecewise expanding maps such that for each $a\in I$
the map $T_a$ admits a unique absolutely continuous invariant probability
measure $\mu_a$. We establish sufficient conditions on such a one-parameter
family such that a given point $x\in[0,1]$ is typical for $\mu_a$ for a full
Lebesgue measure set of parameters $a$, i.e. $$
\frac{1}{n}\sum_{i=0}^{n-1}\delta_{T_a^i(x)}
\overset{\text{weak-}*}{\longrightarrow}\mu_a,\qquad\text{as} n\to\infty, $$
for Lebesgue almost every $a\in I$.