$X$ a random variable, where $-1\leq X\leq \frac{1}{2}, \mathbb E[X]=0$. What is the maximal value of $\mathbb E[X^2]$?

Question

$X$ a random variable, where $$-1\leq X\leq \frac{1}{2}, \mathbb E[X]=0$$

Find the maximal value of $\mathbb E[X^2]$.

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I came across a similar question, asking if the following inequality is always true $$\mathbb E[X^2]\leq \frac{1}{4}$$

The answer is no, for example: $$\mathbb P(X=-1) = \frac{1}{3},\mathbb P(X=\frac{1}{2}) = \frac{2}{3}\implies\\ \mathbb E[X^2]= ((-1)^2)\cdot(\frac{1}{3})+(\frac{1}{2})^2\cdot(\frac{2}{3})=\\ \frac{1}{3} +\frac{2}{3\cdot 4} =\frac{1}{2}$$ I was wondering how can one find the maximum, for any discrete or continuos random variable.

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Remark: I'm not sure this question has a "nice" solution.

Answer 1

For the sake of generality say that $X\in [a,b]$ a.s. (this implies $a\leq 0$). Since $X=\frac{b-X}{b-a}a + \frac{X-a}{b-a}b$ and the square function is convex, $X^2\leq \frac{b-X}{b-a}a^2 + \frac{X-a}{b-a}b^2$ and taking expectations, $$E(X^2)\leq \frac{b}{b-a}a^2 - \frac{a}{b-a}b^2 = -ab$$

This bound is tight: it is attained only for $X\sim \frac{b}{b-a} \delta_{a}+\frac{-a}{b-a} \delta_{b}$.

In the special case where $a=-1$ and $b=\frac 12$ this yields $E(X^2)\leq \frac 12$, with equality iff $X\sim \frac 13 \delta_{-1}+\frac 23 \delta_{1/2}$.

Answer 2

Let \begin{eqnarray*} A &=& \Bbb P(X < 0)\\ B &=& \Bbb P(X > 0)\\ C &=& -\Bbb E(X|X < 0)\cdot A\\ D &=& \Bbb E(X|X > 0)\cdot B\\ E &=& \Bbb E(X^2|X < 0) \cdot A\\ F &=& \Bbb E(X^2|X > 0) \cdot B. \end{eqnarray*} We then have:$$E \leq C \leq A$$$$F \leq \frac12D \leq \frac14B$$$$A + B \leq 1$$$$C = D$$ and we want to maximize $E + F$.

We see that $$1 \geq A + B \geq C + 2D = 3C$$ which implies $C \leq \frac 1 3$, and hence $$E + F \leq C + \frac 1 2 D = \frac 3 2 C \leq \frac 1 2.$$

The equality is obtained when $\Bbb P(X = -1) = \frac1 3$ and $\Bbb P(X = \frac 12) = \frac 2 3$.