Lower bound on Tail Probabilities

Question

Inequalities such as Markov's and Chebyshev’s provide upper bounds on tail probabilities. Are there similar inequalities that give lower bounds in the form $P(X \geq \alpha)>\theta$?

Answer 1

Markov's inequality is also called the first moment method. What you want is the second moment method using bounds for the two first moments to derive the desired inequality :

http://en.wikipedia.org/wiki/Second_moment_method