Fix $\mu >0$ and take $\lambda \geq 0$. Let $B_p \sim \text{Ber}(p)$ with $p = \exp(-\mu - \frac{\lambda}2) $. Define the random variable $Y$ which is Poisson with parameter depending on the value of $B_p$: $$Y \overset{d}= (1 - B_p) \text{Poi}(\mu + \lambda) + B_p \text{Poi}(\mu + \frac \lambda 2)$$ all terms are independent. For any $\lambda$ I would like to find $\epsilon>0$ (possibly depending on $\lambda$) so that $Y$ stochastically dominates a $\text{Poi}(\lambda + \epsilon)$ random variable.
It is possible that this only works for $\mu$ sufficiently large.
