# What is the maximum likelihood estimator for $e^{-\theta}( (P(X_i = 0))$?

Suppose $$X_i$$ is $$iid$$ $$Poisson(\theta)$$

What is the maximum likelihood estimator for $$e^{-\theta}(= P(Xi = 0))$$?

I already found the MLE for the $$\theta$$. how do you then find the MLE of $$e^{-\theta}(= P(Xi = 0))$$ ?

By the functional invariance property of the maximum likelihood estimator, the maximum likelihood estimator of $$e^{-\theta}$$ is just $$\color{blue}{e^{-\widehat{\theta}_{MLE}}}$$, where $$\widehat{\theta}_{MLE}$$ is the maximum likelihood estimator of $$\theta$$.
• so then would that be, the MLE is: $e^{-\hat \theta} = e^{-1/n \sum X_i}$? Since I calculate MLE of $\theta$ is $\hat \theta = 1/n \sum X_i$ – ISuckAtMathPleaseHELPME Mar 6 at 15:36