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$$X \sim \mathcal{N}(0,1)$$ $$Y|X = x \sim \mathcal{N}(x,1)$$

I'm familiar with the normal use of tilde in indicating a random variable follows a distribution (such as in the first line of the attached image), but I haven't run into the tilde as it is used in the second line of the attached image. Is this saying that the random variable is itself the mean? That doesn't make much sense to me.

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This notation means that the distribution of $Y$ conditioned on the event $X=x$ is $\mathcal N(x,1)$.

It is the same as saying $Y|X\sim\mathcal N(X,1)$, although maybe in proper notation: you condition on the event $X=x$ and then use the real number (not random variable) $x$ as the mean for the conditional distribution.

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    $\begingroup$ I think it's helpful if you use parentheses: $(Y|X=x) \sim \mathcal{N}(X, 1)$ $\endgroup$
    – shadowtalker
    Commented Jan 26, 2022 at 6:18
  • $\begingroup$ @shadowtalker This is what I was looking for! Thanks, this makes much more sense. $\endgroup$
    – dcb
    Commented Jan 26, 2022 at 6:53

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