# Chebychev's inequality

I am having a bit of a problem with Chebychev's inequality which is:

$$P(|X-\mu |\geq \alpha )\leq \frac{\mathrm{Var}(X)}{\alpha ^2}$$

For a positive $$\alpha$$. Here X denotes a stochastic variable with mean $$\mu$$ and finite variance. I am asked to give a direct proof of this result, using the inequality

$$Z^2\geq \textbf{1}_{|Z|\geq 1}, \quad Z=\frac{X-\mu}{\alpha}$$.

I have solved this part of the problem already. Next, however, I am asked to give sufficient and necessary conditions for equality in Chebychev, and this is where I could use some help. My initial idea was to use the other inequality, but I am not quite sure how I can translate this into the Chebychev inequality. I figure that equality will be achieved for -1, 0, and 1 respectively, due to the nature of this characteristic function, but I should probably have further conditions on that. Any help would be appreciated.