As you are modeling a non-negative continuous variable, I would suggest the Tweedie distribution. Fundamentally, the Tweedie distribution is considered a family of probability distributions because it incorporates the purely continuous normal and gamma distributions, the purely discrete scaled Poisson distribution, and the class of mixed compound Poisson–gamma distributions which have positive mass at zero, but are otherwise continuous.

Any of these distributions, as well as intermediate cases, can be obtained tweaking some parameters that controls the mean and the variance of the distribution. Some optimizers may help you find the best parameters to be used considering your data.

I hope that helps.