If $f$ is a density function of $\xi$, is there any way to express $\int_{\mathbb{R}}f(x)^2 dx$ in terms of the expected value of $\xi$?

Question

Let $\xi$ be a random variable in $\mathbb{R}$ with probability density function $f$.

The question: Is there any way to express $$\int_{\mathbb{R}}f(x)^2 dx$$ in terms of the expected value of $\xi$? or or is there an upper bound of this expression in terms of the expected value of $\xi$?

Remark: I thank in advance any collaboration from a suggestion to a reference to a book that can help me.

Answer 1

Let $f_a(x) = \frac{1}{2} a I_{[-\frac{1}{a},\frac{1}{a}]}(x)$ for $a > 0$. Then $f_a$ is a probability density for all $a > 0$ and the expected value of the corresponding distribution is always $0$. But

$\int _{\mathbb{R}} f_a(x)^2 dx = \frac{1}{2}a$

which ranges from $(0, \infty)$ for $a > 0$. Thus there is no upper bound in general.

Nevertheless, if $f \in C^1(\mathbb{R})$ and compactly supported in $[0, \infty)$ with $f' \geq - C$ for some $C > 0$, then

$\int _{\mathbb{R}} f^2(x)dx = - \int _{0} ^{\infty} x \cdot 2f(x)f'(x) dx \leq 2C \mathbb{E}[\xi]$

by integration by parts.