My question is the following: if my random variable $X$ has finite or bounded second moment $\mathbb{E}[X^2]\leq B$ can anyone develop any bounds on pdf of $X$.
For example something like this $f(x)\leq g(B)$ where $g(.)$ is function of the bound B.
2026-03-26 20:26:13.1774556773
Probability bounds
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All you can get is $$B \leq \mathbb{E}(X^2) = \int_\Omega x^2 f(x) d\mu(x) \leq \sup_{x\in\Omega} |x|^2 \int_\Omega f(x) d\mu(x) = C_\Omega \Vert f\Vert_{L^1(\Omega; \mu)}$$