# Bound of the variance of a random Variable

I am having trouble trying to prove that given a random variable $Y$ where $0 \lt m_1 \lt Y \lt m_2 < \infty$, where $m_1$ and $m_2$ are constants the

$\displaystyle Var(Y) \le \frac{(m_2 - m_1)^2}{4}$

The question included the following hint to consider $\displaystyle E\left[\left(Y - \frac{m_1 + m_2}{2}\right)^2\right]$

Any help would be greatly appreciated

Thanks Tyler

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Welcome to MSE! Did I get the formatting correct? It helps readability to use MathJax (see FAQ). Regards – Amzoti Apr 25 '13 at 2:13
Yes you did thanks a lot – Tyler Blakeley Apr 25 '13 at 2:14
Several proofs can be found in the answers to this question over on stats.SE. – Dilip Sarwate Apr 25 '13 at 2:53
Thanks Dilip was very helpful – Tyler Blakeley Apr 25 '13 at 3:13

Let $E(Y) = \mu$, then for any $c$ we have $E(Y-c)^2 = E(Y-\mu)^2 + (\mu - c)^2$ so $\text{Var}(Y) =E(Y-\mu)^2 \leq E(Y - c)^2$ for any $c$ in particular $\text{Var}(Y) \leq E(Y - \frac{m_1+m_2}{2})^2$, also $|Y - \frac{m_1 + m_2}{2}| \leq \frac{m_2 - m_1}{2}.$