Question

Please Help me understand this Discrete Math concept.
How does P(X<= a) + P(X > a) = 1 becomes [P(X <= a) >= 1 - E[X]/a ?

Answer 1

Start with Markov's Inequality
$P(x\ \textgreater \ a) \leq \frac{E(x)}{a}$
Given :
$P(x\ \textgreater \ a) +P(x \leq a) = 1 \\ P(x\ \textgreater \ a) = 1-P(x \leq a)$
Substitute into Markov's inequality
$1-P(x \leq a) \leq \frac{E(x)}{a}$
Rearrange:
$P(x \leq a) \geq 1-\frac{E(x)}{a}$