Here is a particular case of Markov's Inequality that I failed to prove:

Let $X$ be a non-negative integer-valued random variable with $\mathbb{E}(X)\leq m$ then $$\mathbb{P}(X=0)\geq 1-m$$

Does anyone has an idea how to prove this? Thank you.


closed as off-topic by user21820, The Phenotype, zz20s, Ethan Bolker, Namaste Feb 22 '18 at 18:47

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$$m\geq\mathsf EX=\sum_{n=0}^{\infty}n\mathsf P(X=n)\geq\sum_{n=1}^{\infty}\mathsf P(X=n)=\mathsf P(X\geq1)=1-\mathsf P(X=0)$$

  • $\begingroup$ Hence it is not a corollary to Markov's Inequality. $\endgroup$ – user448839 Feb 22 '18 at 11:03
  • $\begingroup$ To find it as corollary of Markovs inequality subtitute $a=1$ in $aP(X\geq a)\leq EX$ and work out. $\endgroup$ – drhab Feb 22 '18 at 12:05

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