Suppose that Y and X are two random variables.
What is the difference between $Y|X$ and $Y|X=x$?
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3$\begingroup$ Neither are defined as such. When writing $\mathbb E[Y|X]$ this is understood as a random variable that is a (deterministic) transform of the random variable $X$. When writing $\mathbb E[Y|X=x]$ this is understood as the realisation of the above rv when the realisation of $X$ is $x$. $\endgroup$– Xi'anCommented Dec 7, 2021 at 20:10
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$\begingroup$ If neither are defined as such then how can we say that $Y|X=x\sim \mathcal{N}(5x,\,\sigma^{2})\,$. Am I missing something? $\endgroup$– gnikolCommented Dec 7, 2021 at 21:00
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1$\begingroup$ This means that the $Y$ random variable, conditioned on the realization $x$ of the random variable $X$ is normally distributed with mean $5x$ and standard deviation $\sigma$. $\endgroup$– Guilherme MartheCommented Dec 7, 2021 at 21:17
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1$\begingroup$ $Y|X$ as such does not have a meaning. $Y$ is a random variable and $(X,Y)$ is a pair of random variables. $\endgroup$– Xi'anCommented Dec 8, 2021 at 7:20
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1$\begingroup$ See also the discussion on this question: stats.stackexchange.com/q/507236/7224 $\endgroup$– Xi'anCommented Dec 8, 2021 at 7:43
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