In order to investigate the size of the cluster of a given vetex in a random graph I need to use a fact about stochastic dominance that I don't know how to prove.
Namely, I am looking for a proof of the fact that $\text{Binom}(n, p)$ is stochastically dominated by $\text{Poi}(\lambda)$, where $\lambda=p(n+1)$ and $p\leq\frac{1}{n+1}.$
By stochastic dominace I mean the following:
$\mu$ is stochastically dominated by $\nu$ if $\mu(A)\leq\nu(A)$ for any upward closed event $A \subseteq \mathcal{P}$ (i.e. $x \in A$ and $x \leq y \implies y\in A$).
Thank you for your help in advance.