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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 $P(X \geq \alpha)>\theta$?

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up vote 3 down vote accepted

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

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