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For random piecewise linear systems T of the interval that are expanding on average we construct explicitly the density functions of absolutely continuous T-invariant measures. If the random system uses only expanding maps our procedure produces all invariant densities of the system. Examples include random tent maps, random W-shaped maps, random 
$\beta $-transformations and random Lüroth maps with a hole.
$\unicode[STIX]{x1D6FD}$
-transformation with a hole at 0
For 
$\unicode[STIX]{x1D6FD}\in (1,2]$
the 
$\unicode[STIX]{x1D6FD}$
-transformation 
$T_{\unicode[STIX]{x1D6FD}}:[0,1)\rightarrow [0,1)$
is defined by 
$T_{\unicode[STIX]{x1D6FD}}(x)=\unicode[STIX]{x1D6FD}x\hspace{0.6em}({\rm mod}\hspace{0.2em}1)$
. For 
$t\in [0,1)$
let 
$K_{\unicode[STIX]{x1D6FD}}(t)$
be the survivor set of 
$T_{\unicode[STIX]{x1D6FD}}$
with hole 
$(0,t)$
given by In this paper we characterize the bifurcation set 
$$\begin{eqnarray}K_{\unicode[STIX]{x1D6FD}}(t):=\{x\in [0,1):T_{\unicode[STIX]{x1D6FD}}^{n}(x)\not \in (0,t)\text{ for all }n\geq 0\}.\end{eqnarray}$$
$E_{\unicode[STIX]{x1D6FD}}$
of all parameters 
$t\in [0,1)$
for which the set-valued function 
$t\mapsto K_{\unicode[STIX]{x1D6FD}}(t)$
is not locally constant. We show that 
$E_{\unicode[STIX]{x1D6FD}}$
is a Lebesgue null set of full Hausdorff dimension for all 
$\unicode[STIX]{x1D6FD}\in (1,2)$
. We prove that for Lebesgue almost every 
$\unicode[STIX]{x1D6FD}\in (1,2)$
the bifurcation set 
$E_{\unicode[STIX]{x1D6FD}}$
contains infinitely many isolated points and infinitely many accumulation points arbitrarily close to zero. On the other hand, we show that the set of 
$\unicode[STIX]{x1D6FD}\in (1,2)$
for which 
$E_{\unicode[STIX]{x1D6FD}}$
contains no isolated points has zero Hausdorff dimension. These results contrast with the situation for 
$E_{2}$
, the bifurcation set of the doubling map. Finally, we give for each 
$\unicode[STIX]{x1D6FD}\in (1,2)$
a lower and an upper bound for the value 
$\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FD}}$
such that the Hausdorff dimension of 
$K_{\unicode[STIX]{x1D6FD}}(t)$
is positive if and only if 
$t<\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FD}}$
. We show that 
$\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FD}}\leq 1-(1/\unicode[STIX]{x1D6FD})$
for all 
$\unicode[STIX]{x1D6FD}\in (1,2)$
.
$\beta $
-transformations
We compare a piecewise linear map with constant slope 
$\beta \gt 1$
and a piecewise linear map with constant slope 
$-\beta $
. These maps are called the positive and negative 
$\beta $
-transformations. We show that for a certain set of 
$\beta $
s, the multinacci numbers, there exists a measurable isomorphism between these two maps. We further show that for all other values of 
$\beta $
between 1 and 2 the two maps cannot be isomorphic.
