MLE of P(X<2) - Exponential distribution

[SandMan249]

Homework Statement

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

Homework 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!

 

[Ray Vickson]

Homework Statement

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

Homework 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!

Is the following statement of your problem correct? You observe n independent values of X and observe that m of them have {X < 2} (or {X <= 2}). From that, you want to estimate θ = P{X <= 2}. If that is truly the statement, what does the exponential distribution have to do with it? (Of course, if you want to estimate λ you need to know the distribution, but that is not what you said you want to estimate.)

RGV