Poisson approximation of $X$ by $Poisson(E[X])$

Question

I've tried to find something, but couldn't find anything about the following question.

Is it possible to approximate any random variable $X$ with $E[X]=o(1)$ by a Poisson random variable $Poisson(E[X])$.

Especially, I would like to Show that

$$Pr[X \geq x] \sim Pr[Poisson(E[X])\geq x]$$

for some $x=O(1)$.

Do you know results on the (relative and absolute) error of this Approximation?

EDIT: Would it help, if we knew that $X$ is a sum of Binomial random variables? Thus $X_i \sim Bin(n_i, p_i)$ and $X=\sum_{i} X_i$ and $E[X]=\sum_i n_i p_i=o(1)$. Can we then approximate $X$ by $Poisson(\sum_i n_i p_i)$?

Answer 1

Note that the poisson distribution with parameter $\lambda$ is the limit case ($n\to \infty$) of the sum of $n$ indepent Bernouilli trials with parameter $\frac \lambda n$. Thus, if the binomials have a large coefficient, we may approximate them with a certain poisson distribution, and then we can just replace the sum of the poisson distributions by a single poisson distribution.