MLE of Bernoulli trials

1. Feb 24, 2012

SandMan249

1. The problem statement, all variables and given/known data
Two independent bernoulli trials resulted in one failure and one success. What is the MLE of the probability of success θ is it is know that θ is at most 1/4

2. Relevant equations
f(x,θ) = θx (1-θ)1-x

3. The attempt at a solution
Now, I know how to find the likelihood and use it to solve for the MLE. But I am not sure how the "θ is at most 1/4" would factor into the equation.

For a Bernoulli trial: f(x,θ) = θx (1-θ)1-x
L(θ) = θ(1-θ)
L'(θ) = 1-2θ ----> equate to zero
θ(hat) = 1/2 (which is the MLE)

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

2. Relevant equations

3. The attempt at a solution

2. Feb 25, 2012

Office_Shredder

Staff Emeritus
You're trying to maximize L(θ) on the interval [0,1/4]. Recall a basic calculus fact that the maximum of a function on a closed interval is either a critical point (where the derivative is zero) or an end point

3. Feb 25, 2012

SandMan249

Thank you.
In this case, since the function is simply: θ(hat) = 1/2 (a constant)
The maximum will lie at 1/4
Correct?

4. Feb 25, 2012

Office_Shredder

Staff Emeritus
The critical point is at 1/2. But the function you want to maximize is L(θ) = θ(1-θ)