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. 094160Q:

Conditional expectation of the product of a non-negative continuous random variable and a bounded random variable

If $X$ is non-negative, continuous random variable and $\mathbb{E}[X^k]a)=0$, $\forall a\in\mathbb{R}$, but the definition of conditional expectation gives an upper bound for $|\mathbb{E}[X\,|\, X\le1]|$, which is not the case. I cannot find a way to simplify the problem.

A:

You can write
$$\mathbb{E}[X\mid X\le 1] = \frac{\mathbb{E}[X^2\mid X\le 1]}{\mathbb{E}[X\mid X\le 1]}$$
since $X^2$ is non-negative, and so you can use the tower property of conditional expectation (and $0\le 1$) to derive
$$\mathbb{E}[X\mid X\le 1] \le \frac{\mathbb{E}[X]}{\mathbb{E}[1\mid X\le 1]}$$
and by the linearity of expectation this is at most $\mathbb{E}[X]/\mathbb{E}[1]$.

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