Conditional distribution distributed as notation

Question

What's the proper way to write:

$$(X \mid \mu = t) \sim \mathcal{N} (t, 1)$$

Some people write it as $X|\mu \sim \mathcal{N}(\mu, 1)$, however I find this confusing as it isn't clear what is a random variable and what is a constant here.

Answer 1

I would define an upper case $M$ to be the random expected value and then write

$$ (X|M=\mu)\sim \mathcal{N}(\mu,1) $$

or

$$ X|_M \sim \mathcal{N}(M,1) $$

It is essentially the same that you wrote, only that the convention of using upper case letters for random variables helps.

Answer 2

In this case it's pretty simple, you can define $Y \sim \mathcal{N} (0, 1)$ and then:

$$ X = \mu + Y$$

or

$$ X = \mu + \mathcal{N} (0, 1)$$