I need help to prove the following lower bound for tail probability. I have tried using well-known inequalities like Chebyshev and Paley-Zygmund, but cannot get the required bound.
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space, and $X$ be a random variable with $\mathbb{P}(X\in [-M,M])=1$ for some positive constant $M$. Show that $\mathbb{P}\left(|X-\mu|\geq \frac{\sigma^2}{4M} \right)\geq \frac{\sigma^2}{8M^2}$, where $\mu= \mathbb{E}(X)$ and $\sigma^2=\text{Var}(X)$.
Thank you.
Additional question: How can I also prove that $\mathbb{P}\left(X\geq \mu+ \frac{\sigma^2}{4M} \right)\geq \frac{\sigma^2}{8M^2}$ ?