MLE of P(X<2) - Exponential distribution

#1
The problem statement:
Find the MLE of θ = P (X≤ 2) in a random sample of size n selected from an exponential distribution EXP(λ)

Relevant equations

f(x, λ) = λ e^(-λx)
F(x, λ) = 1 - e^(-λx)

The attempt at a solution
I know how to find the MLE of the mean of an exponential distribution. But I am not sure how I can tackle this problem.

We know that P ( X≤ 2) = ∫f(x) 0,2 = F(4)

How do I get to the Likelihood from here?

Thanks!
 

BGM

TS Contributor
#2
If \( \theta = \Pr\{X \leq 2\} = g(\lambda) \), then by the functional invariance of MLE, the MLE of \( \theta \) is \( \hat{\theta} = g(\hat{\lambda}) \) where \( \hat{\lambda} \) is the MLE of \( \lambda \). This is particularly easy to see if \( g \) is a bijective function.