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Let $X$ be a non-negative random variable bounded on $[0,1]$. Is it true that $\mathbb{E}[X^2]\leq k \mathbb{E}[X]^2$ for some constant $k$? If not, are there any minimal assumptions on $X$ where this holds?

It's easy to show that this holds for Bernoulli random variables, but I am not sure they are the worst examples.

A related question is https://stats.stackexchange.com/questions/617592/expectation-of-first-of-moment-of-symmetric-r-v-in-terms-of-variance, which shows the statement is false for general $X$'s (that are not non-negative).

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Despite your "It's easy to show ...", Bernoulli random variables with $\mathbb P(X=1)=p$ are a counter-example: $$\dfrac{\mathbb{E}[X^2]}{ \mathbb{E}[X]^2} = \dfrac{p}{p^2} = \dfrac1p$$ and that is unbounded as $p \to 0$.

If there were some $\delta>0$ with $\mathbb{P}(X \in [\delta,1])=1$, then there would be a finite upper bound on $\dfrac{\mathbb{E}[X^2]}{ \mathbb{E}[X]^2}$, and clearly there is a lower bound of $1$.

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  • $\begingroup$ With my $\mathbb{P}(X \in [\delta,1])=1$ suggestion, $k=\frac{1}{\delta^2}$ is certainly an upper bound since $\mathbb{E}[X^2]\le 1$ and $\mathbb{E}[X]^2\ge \delta^2$, and I suspect $k =\frac{(1+\delta)^2}{4\delta}$ may also be an upper bound. $\endgroup$
    – Henry
    Apr 12 at 23:58

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