have a little trouble here with understanding one thing.
Let us assume $X,Y,Z$ three random variables $X$ is independent of $Z$. My question is if the following holds:
$$\mathbb E[XY|Z]=\mathbb E[X]\cdot \mathbb E[Y|Z]$$
2026-04-01 15:40:33.1775058033
probability theory: question about conditional means
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No, this is not true in general. Suppose $X = Y$. Then since $X$ and $Y$ are independent of $Z$ we have $\mathbb{E}[XY|Z] = \mathbb{E}[X^2]$ but $\mathbb{E}[X]\mathbb{E}[Y|Z] = \mathbb{E}[X]^2$ so $\mathbb{E}[XY|Z] \ne \mathbb{E}[X]\mathbb{E}[Y|Z]$ unless $X$ is constant.