We consider a simple example of a one-parameter family of random maps in the interval that exhibits a phase transition phenomenon in the sense of a spontaneous transition from non-existence to existence of an absolutely continuous invariant measure by changing the parameter. For this example of random maps, we estimate numerically the critical value at which the so-called transition occurs. This is done through the numerical computation of its invariant densities and the Lyapunov exponent. We further investigate the behavior of the rate of decay of correlations for the different values of the parameter, changing from power-law-like to exponential-like behavior. We also study a family of random maps with a non-expansive branch having no phase transition. For this family of random maps, we compute the invariant densities for all values of the parameter.