# MLE of P(X<2) - Exponential distribution

1. Apr 21, 2012

### SandMan249

1. The problem statement, all variables and given/known data

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

2. Relevant equations

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

3. 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!
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Apr 21, 2012

### Ray Vickson

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

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