If $X$ and $Y$ are random variables, does $\mathbb{E}[{X+Y|X}] = X + \mathbb{E}[Y]$? Note: The answer is still a random variable.$\\$ I should add that I derived this law from the following statements: $$\mathbb{E}[{X+Y|X}] = \mathbb{E}[{X|X}] + \mathbb{E}[{Y|X}] \\ =X + \mathbb{E}[{Y|X}] \\ = X + \mathbb{E}[Y]$$
2026-03-31 03:45:26.1774928726
If $X$ and $Y$ are random variables, does $\mathbb{E}[{X+Y|X}] = X + \mathbb{E}[Y]$?
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No. Consider, for instance, if $X$ is any random variable and $Y=X$.