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Assume $\Omega = \mathbb{N}_0$ and $k > 0$. Prove that $f(\omega) = e^{-\lambda} \cdot \frac{\lambda^{\omega}}{\omega !}$ is a mass probability function. Showing $f \geq 0$ is trivial as well as $\sum \limits_{\omega \in \Omega} f(\omega) = 1$. I just can't seem to figure out how to prove $f(\omega) \leq 1$ except for maybe showing that the limes for a fixed $\lambda$ and $\omega$, respectively, never surpasses 1. But that doesn't seem very subtle. Is there a more elaborate way of showing this?

user234723
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1 Answers1

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Note that for $\lambda>0$, $\frac{\lambda^{\omega}}{\omega! }<e^{\lambda}$ by Taylor expansion. Multiply by $e^{-\lambda}$ and you have your claim.