In this paper we revisit some issues related to the use of the Box-Cox transformation in censored and truncated regression models, which have been overlooked by the econometric and statistical literature. We first analyze the shape of the density function of the random variable which, rescaled by a Box-Cox transformation, leads to a normal random variable. Then, we identify the value ranges of the Box-Cox scale parameter for which a regular expectation of the derived random variable does not exist. This result calls for an extension of the concept of expectation, which can be computed regardless of the value of the scale parameter. For this purpose, we extend the concept of mean of a rescaled series of observations to the case of a random variable. Finally, we run estimates of censored and truncated Box-Cox standard Tobit models to determine the range of the scale parameter most relevant for empirical demand analyzes. These estimates highlight significant deviations from the assumption of normality of the dependent variable towards highly right skewed and leptokurtic distributions with no expectation.