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I am slightly confused about the different between

$\mathrm{E}[Y|X = x]$

and

$\mathrm{E}[Y|X]$

and similarly for Variance.

It seems to me the first should be a scalar, because we first pick a specific $X = x$ and then get the expected value of $Y$ within that set whereas the second one is a random variable that depends on the random variable $X$. Is that correct?

Any definition using the probabilities $\mathrm{P}(X)$, $\mathrm{P}(Y)$, $\mathrm{P}(Y|X)$ and $\mathrm{P}(Y, X)$ is appreciated.

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    Your interpretation is correct. $\Bbb E[Y|X]$ is a random variable whose value at $X=x$ is $\Bbb E[Y|X=x]$. $\Bbb E[Y|X]$ is called the conditional expectation of $Y$ given $X$. A similar statement holds for the conditional variance of $Y$ given $X$ (symbolized by $\text{Var}[Y|X]$).2012-02-19
  • 0
    You need to be clear whether $X$ and $Y$ are random variables or events. If they are random variables then $E[Y|X]$ may be a function of a random variable and so a random variable, but then so too is $P(X)$ which may not be what you intend as you describe it as a probability.2012-02-19

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