Bound of the variance of a random Variable

Question

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

Answer 1

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}.$