In this paper, a new family of continuous random variables with positive support is introduced. Its density function has the capacity to incorporate features of unimodality and bimodality. Special attention is paid to the lognormal distribution which is included as a particular case. Its density function is given in closed-form, allowing probabilities, moments and other related measures such as skewness and kurtosis coefficients to be computed easily. In addition, a stochastic representation of the family that enables us to generate random variates of this model is also presented. Some properties related with the right tail and actuarial aspects of the distribution are also shown. This new family of distributions is numerically illustrated with data taken from the Medical Expenditure Panel Survey (MEPS), conducted by the US Agency of Health Research and Quality and with a well-known data set which has been studied widely in the actuarial literature.