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Probability P(X=k) in Poisson discrete probability distribution is given by:

P(X=k)=\frac{e^{-\lambda}\ \lambda^k}{k!}\\\\Given\ \frac{P(X=3)}{P(X=2)} \ \textgreater \ 1\\\\so\ \frac{\lambda^3*2!}{3!*\lambda^2}\ \textgreater \ 1\\\\.Hence,\ \lambda\ \textgreater \ 3

So the minimum value of the mean is 3.

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