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Suppose $E(2^{X})=4$. Prove that $P(X\geq3)\leq1/2$.

Using Jensen's inequality we know $E(X)\leq2$ but since $X$ is a general random variable we can't use Markov's inequality to get a bound for $P(X\geq3).$ Any suggestions?

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HINT 1: Apply Markov's inequality to the nonnegative random variable $Y = 2^X$, rather than directly to $X$.

HINT 2: This obvious statement might come in handy: $X \geqslant 3 \iff 2^X \geqslant 8$.