MLE of P(X<2) - Exponential distribution

[sandeep249]

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]

If , then by the functional invariance of MLE, the MLE of is where is the MLE of . This is particularly easy to see if is a bijective function.

[sandeep249]

Thanks, that was very helpful!