Default positive/(non-negative) probability distribution

Solution 1:

The normal distribution is often used to model phenomena that give strictly positive results. Example: Heights of women in a particular population with mean $\mu = 67$ inches and standard deviation (SD) $\sigma = 3.5.$

Technically, using a normal is 'wrong' because a normal distribution has a left tail that extends to negative infinity, and obviously no women have negative heights. However, almost all probability is contained within 3 or 4 SD of the mean, so zero (almost 20 SD below) plays no practical role.

Similarly, SAT (and other) exam scores are sometimes described as normal, totally ignoring the minuscule probability that would correspond to nonexistent negative scores.

The gamma family of distributions places all its probability on the positive half-line. Gamma distributions are right-skewed with mode < median < mean. The shape is governed by a shape parameter.

For larger shape parameters the distribution has a left sided tail and a somewhat more pronounced right sided tail. See graphs of various gamma densities in the Wikipedia article. (A scale or rate parameter helps to determine the variance of a gamma distribution.)

Gamma distributions are used to model waiting times, and many other phenomena in social, biological, and physical sciences. The gamma family of distributions is not as widely used as the normal family, but if any family of continuous distributions can be described as the 'default' non-negative family, the gamma family would be the prime candidate.

In queueing theory (concerned with the behavior of waiting lines), gamma distributions with integer shape parameters are called Erlang distributions. The chi-squared family is another sub-family of gamma.

Exponential distributions are members of the gamma family with shape parameter 1; strongly skewed with no left-sided tail.

A few other particularly useful continuous non-negative families, among many, are the lognormal (taking logs of data produces normality), Weibull, Rayleigh and Pareto. (Also discussed in Wikipedia articles.)