# Poisson distribution/ finding special k

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?

• – StubbornAtom Apr 28 at 15:38
• Thank you for your answer:) – Steven33 Apr 28 at 16:01