Correct notation for random variable with probability distribution

Question

For a scientific publication I am documenting the algorithm of my computer model. The model uses the Monte Carlo method and has some random variables input. Now I would like to document that correctly.

So for instance variable $a$ is determined by multiplying variable $b$ with a random number $X$, which is drawn from a log-normal probability density function. My guess for a notation would be as follows:

$a_{i} = b_{i} \cdot X_{i}, \quad X_{i} \sim \ln \mathcal{N}(\mu_{i}, \sigma_{i}^{2})$

But I ask myself:

1. Is this $\LaTeX$ notation correct (use of comma, \quad, $\ln\mathcal{N}$) as an equation in a publication?
2. What is a typical random variable symbol? $X$ seems to be quite common.
3. How can I document other distributions, such as Weibull, log-logistic, discrete uniform?

Answer 1

There do not seem to be standard abbreviations for various distributions. Just be sure to write the name out fully before starting to use an abbreviation.

Also for normal be sure to say whether you are using SD $\sigma$ or variance $\sigma^2$ for a dispersion parameter, and for exponential distribution whether you are using mean $\mu$ or rate $\lambda$ for the parameter.

In the US is it common to use capital Latin letters near the end of the alphabet $U, V, W, X,$ etc. for random variables, small Greek letters for parameters of distributions (as above), small Latin letters for constants, and boldface for vectors and matrices. But there are many exceptions. Various sub-disciplines have their own considerable variations.

It is possible that a journal will enforce a particular style. (Maybe it's best to look at several published papers in target journals to see what is common usage there. Or at published submission guidelines for authors.)

I would not be surprised to see Answers or Comments here with a variety of different opinions.