What would be an accurate mathematical notation for the following:
A sequence of $T$ random variables, where each element up to $\kappa$ is a normally distributed with mean $\mu_1$ and variance $\sigma^2_1$, and each element from $\kappa$ to $T$ is a normally distributed with mean $\mu_2$ and variance $\sigma^2_2$.
I came up with something like:
$X = \{x_t:t=1,\dots,\kappa,\dots T\}$ where $x_t \sim \operatorname{N}(\mu_1,\sigma^2_1)$ if $t \leq \kappa$ and $x_t \sim \operatorname{N}(\mu_2,\sigma^2_2)$ if $t> \kappa$.
But I have no idea if that is correct. Does anyone know an elegant way to write this? Thanks in advance!