I have $X_1, X_2, . . . , X_n$ are i.i.d. observations from a Poisson distribution with
$$f_{\lambda}x = \frac{\lambda ^ x \exp^{-\lambda}} {x!}$$ for $x = 1,2,...,n.$
I have found the MLE of $\lambda$ : $\hat \lambda = \bar X.$
I am then asked to
1) Find the MLE of the probability that a new observation from the underlying distribution is equal to 1 by using the invariance property of the MLE.
2) Find the MLE of the probability that a new observation from the underlying distribution is greater than or equal to $1$.
How can I solve these two?
The invariance property of the MLE says that if $\hat \theta$ is an MLE of $\theta$, then $f(\hat\theta)$ is an MLE of $f(\theta)$ if $f$ is a one-to-one function.
If $X \sim \operatorname{Poisson}(\lambda)$, then what is $\Pr[X = 1]$ as a function of the unknown parameter $\lambda$? Is this function one-to-one?
What is $\Pr[X \ge 1]$? Is this function also one-to-one?