# Correct notation for random variable with probability distribution

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:

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

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?

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.