Question: Suppose \(X \sim\) Poisson \((\lambda)\) and \(\alpha \in(0,1)\).

1. Apply the Markov inequality for the random variable \(Y_{s}=e^{-s X}\) to get an upper bound for \(P(X<\alpha \lambda)\).
2. Minimize the bound obtained in (a) with respect to \(s\) to have a sharper bound.

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