maximum-likelihood estimation for P (X > λ) when xi ~ pois ( λ)

Question

Suppose X1, · · · , Xn are i.i.d. samples following Pois(λ) distribution.
X is a random variable, X ∼ Pois(λ).
I need help in finding the maximum-likelihood estimation for ρ = P (X > λ).

I tried to find the likelihood function, knowing the poisson distribution formula, but I'm not sure how to address the condition of X > λ.
Chernoff's bound / Markov's inequality cannot be used, because inequality isn't suitable for finding the likelihood function.

Thanks

Answer 1

It is well$-$known that MLE for $\lambda$ is the sample mean $\bar{X}$. Moreover, $$\rho=1-\sum_{k=0}^{\lfloor{\lambda}\rfloor}\frac{e^{-\lambda}\lambda^k}{k!}$$ By the invariance property of the MLE, we see that the MLE of $\rho$ is $$\rho=1-\sum_{k=0}^{\lfloor{\bar{X}}\rfloor}\frac{e^{-\bar{X}}{\bar{X}}^k}{k!}$$