Timeline for What's the difference between Y|X and Y|X=x

8 events

when toggle format	what		by	license	comment
Dec 8, 2021 at 7:43	comment	added	Xi'an		See also the discussion on this question: stats.stackexchange.com/q/507236/7224
Dec 8, 2021 at 7:20	comment	added	Xi'an		$Y\|X$ as such does not have a meaning. $Y$ is a random variable and $(X,Y)$ is a pair of random variables.
Dec 7, 2021 at 22:51	comment	added	gnikol		So $Y\|X=x$ is defined as a conditional probability distribution conditional on the realization of the random variable $X$. Also, as Xi'an correctly points out $E[Y\|X] $ is also a random variable and $E[Y\|X=x] $ is a function of $X$. My question is how $Y\|X$ is defined?
Dec 7, 2021 at 21:17	comment	added	Guilherme Marthe		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 $\sigma$.
Dec 7, 2021 at 21:00	comment	added	gnikol		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?
Dec 7, 2021 at 20:59	history	edited	gnikol	CC BY-SA 4.0	added 1 character in body
Dec 7, 2021 at 20:10	comment	added	Xi'an		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 $x$.
Dec 7, 2021 at 20:06	history	asked	gnikol	CC BY-SA 4.0