Let $X$ be a discrete random variable with its probability mass function:

$$p_X(k)= P(X=k) \ for\ k\in \mathbb{N}_0 $$.

I want to find $k_0$,such that $p_X(k_0) \geq p_X(k) \forall k \in \mathbb{N}_0 $ I consider the Poisson distribution.

My idea ist to consider $ \frac{\frac{e^{-\lambda} \lambda^{k+1}}{(k+1)!}}{\frac{e^{-\lambda} \lambda^{k}}{k!}} = \frac{\lambda}{k+1}>1 \Leftrightarrow \lambda >k+1 \Leftrightarrow \lambda-1 >k$

So $ k_0 = \lambda-1 \ \ for \lambda \in \mathbb{N} $

Is this the right track?


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