Let $x_1,\ldots,x_n$ are observations of iid Poisson$(\lambda)$ random variables. Suppose that the Bernoulli random variable $Y$ is derived from $X$ by
$$ Y = \begin{cases} 1, & \text{if $X=0$} \\ 0, & \text{if $X>0$}. \end{cases}$$
Does it mean $\mathbb P(Y_i=1)=\mathbb P(X_i=0)=\frac{\exp(-\lambda)\lambda^0}{0!}=\exp(-\lambda)?$
That is, exponent of negative value of rate parameter of a Poisson distribution is the probability of success of a Bernoulli distribution???