Or you may have to instead declare their joint distribution - that depends on what purpose you have for considering the random variables. If they are not independent, you can say that $X_i, i=1,2,\dots,n$ are possibly dependent, but (marginally) identically distributed as $N(\mu,\sigma^2)$. for general parameters $\ N(\mu,\sigma^2),i=1,2,\dots n$, given that you have $n$ variables (iid stands for independent and identically distributed). Dilip's answer also gives a nice account of what other possible interpretations there are when the notation is less clear than $\sigma^2$, e.g. Other answers already tell you what the notation means, namely that $X$ is a normally distributed random variable with some mean $\mu$ and variance $\sigma^2$. How is the notation $X \sim N(\mu,\sigma^2)$ read? What follows is my attempt at a more to the point response. It is my (possibly mistaken) impression that in Engineering circles one sees more often $N(\mu, \sigma)$ (which conforms with the general notational rule), while in Econometrics circles almost always one sees $N(\mu, \sigma^2)$ (which falls to the temptation of providing the moments, by treating $\sigma^2$ as the base parameter and not as the square of it).ĮDIT: My previous answer failed to answer the actual question.
In the case of the normal distribution, the parameter $\mu$ happens to also be the mean of the distribution, while the parameter $\sigma$ happens to be the square root of the variance. The mean is $k\theta$ and the variance $k\theta^2$.
For a Gamma (shape-scale parametrization), we write $G(k,\theta)$. The mean of the distribution is $(a+b)/2$ while the variance is $(b-a)^2/12$. So for a Uniform that ranges in $$ we write $U(a,b)$. We do not note down the moments, which usually are a function of, but not equal to, these parameters. This is how the symbols are used at least in Statistics/Econometrics.Īs regards the notational conventions for a distribution, the normal is a borderline case: we usually write the defining parameters of a distribution alongside its symbol, the parameters that will permit one to write correctly its Cumulative distribution function and its probability density/mass function.
As regards the use of symbols $\sim$ ("follows", "is distributed according to "), and $\approx$ ("equals approximately"), see this answer.