What is the correct notation if two random variables belong to the same distribution? I want to explain that both the real and imaginary part of a complex variable follow a zero-mean complex Gaussian distribution. How can I write that?
For one variable I'd write
$a \sim \mathcal{N}(0,\sigma^2)$.
But for two? Maybe
$\{a,b\} \sim \mathcal{N}(0,\sigma^2)$?
What's the correct notation?
 A: I've never seen a notational standard for this in any of my courses.  As long as you define your notation or make it really obvious what you mean, I think you'll be fine.  I personally write it as $$x, y \sim \mathcal{N}(0,\sigma^2)$$
and haven't had any problems with my professors.
A: Wikipedia has an answer: http://en.wikipedia.org/wiki/Random_variable#Equality_in_distribution
They notate the fact that two random variables $X,Y$ are "equal in distribution" by $X \stackrel{d}{=} Y$.
I have no idea how common this notation is. Also note that the mere fact could also be called identically distributed, as in "i.i.d.".
A: You could write
$$ (a,b)^T \sim \mathcal{N}_2(\mathbf{0},\sigma^2\mathbf{I}) $$
to indicate that the vector of random variables $\begin{bmatrix} a\\b\end{bmatrix} $ belongs to the multivariate normal distribution of dimension 2 with zero mean vector $\mathbf{0}$, and where $a$ and $b$ are independent and each have variance $\sigma^2$.
This seems a bit indirect though. You might be better off saying it in words as per http://everything2.com/title/complex+Gaussian+distribution. I.e. something like "the complex number $a+ib$ follows a two-dimensional uncorrelated Gaussian distribution with mean $0$ and variance per real dimension of $\sigma^2$".
