Strictly positive random variables Suppose $X\sim N(\mu, \sigma^{2})$ with some small $\sigma^{2}$ and largish $\mu$. Now $X$ will be rarely negative.
Suppose I need random variables that are strictly positive but otherwise normal-like. Have this kind of variables been studied before? It is not something that I find from my statistics books..
 A: A truncated normal distribution might fit the bill. (Or a better statistics book.) The truncated normal is obtained by discarding whatever is below zero, in your situation. The pdf, cdf and the moments are fully described in the linked Wikipedia article.
A: Depending on how "normal-like" you want your variable, you might consider a log-normal distribution, in which the logarithm of the variable has a normal distribution. The variable itself is thus always non-negative, and for the type of distribution you specify (large mean, small variance) the variable might look close to normal itself.
For measurements on items than are necessarily non-negative, a log-normal distribution can be more appropriate than a normal distribution if the measurement error is proportional to the value measured. If you try to model such measurements with a normal distribution, you can get into trouble because the variance isn't constant over the range of measurements. On the log scale, the variance would tend to be independent of the measured values.
