Using chernoff bounds for poisson distribution

Question

Chernoff inequality can be easily used for sum of independent Bernoulli distributions $X = X_1+X_2+...+X_n$:

$P(X \geq (1+ \varepsilon)\mu) \leq e^{-\frac{\varepsilon^2\mu}{3}}$, where $\mu = EX$

How can we use it to estimate upper bound for $P(Y \geq (1+ \varepsilon)\lambda)$, where Y is random variable with Poisson distribution with parameter $\lambda$?

Since it is all about estimations, as far as I understand, we can approximate Poisson distribution with binomial one, but still it is far from Bernoulli distribution which can be used here.

Answer 1

For large enough $n$, the Poisson distribution $\mathsf{Poisson}(\lambda)$ can be approximated as $\mathsf{Binomial}(n, \frac{\lambda}{n})$. Recall that a random variable $X \sim \mathsf{Binomial}(n, \frac{\lambda}{n})$ can be represented as a sum of $n$ Bernoulli random variables; that is, $$ X = X_1 + X_2 + \cdots + X_n $$ where for $\forall 1 \leq i \leq n$, $$ X_i = \begin{cases} 1 & \text{with probability } \frac{\lambda}{n} \\ 0 & \text{with probability } 1 - \frac{\lambda}{n} \end{cases} $$ You can start from here and obtain an estimation.