TITLE: Poisson approximation of $X$ by $Poisson(E[X])$
QUESTION [1 upvotes]: 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 $Poisson(\sum_i n_i p_i)$?

REPLY [0 votes]: 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.
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