  This paper deals with the randomized heat equation defined on a general
bounded interval $[L_1,L_2]$ and with non-homogeneous boundary conditions. The
solution is a stochastic process that can be related, via changes of variable,
with the solution stochastic process of the random heat equation defined on
$[0,1]$ with homogeneous boundary conditions. Results in the extant literature
establish conditions under which the probability density function of the
solution process to the random heat equation on $[0,1]$ with homogeneous
boundary conditions can be approximated. Via the changes of variable and the
Random Variable Transformation technique, we set mild conditions under which
the probability density function of the solution process to the random heat
equation on a general bounded interval $[L_1,L_2]$ and with non-homogeneous
boundary conditions can be approximated uniformly or pointwise. Furthermore, we
provide sufficient conditions in order that the expectation and the variance of
the solution stochastic process can be computed from the proposed
approximations of the probability density function. Numerical examples are
performed in the case that the initial condition process has a certain
Karhunen-Lo\`eve expansion, being Gaussian and non-Gaussian.
